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As with any technology, there’s knowing Rails and then there’s really knowing Rails. This guide offers a sampling of questions that are key to evaluating the breadth and *mapp vs. ohio* depth of a candidate’s mastery of the framework.
Just as France’s Train a Grande Vitesse (TGV) (traveling at **died** speeds of up to teenage cliques 320 km/h) dramatically reduces travel time for modern day rail passengers, Ruby on Rails (a.k.a. “Rails”) substantially reduces the time and *did garrett morgan* effort required to build powerful web applications. Tim O’Reilly (founder of **for Premature Ejaculation** O’Reilly Media) refers to when did garrett Rails as breakthrough technology and Gartner Research noted in a recent study that many high-profile companies are using Rails to build agile, scalable web applications.
The rate at **warhol** which Rails has gained popularity is noteworthy, with estimates of over 200,000 web sites currently built with the **did garrett died**, technology. Today, many high-profile companies are using Rails to build agile, scalable web applications.

Examples include Twitter, GitHub, Yammer, Scribd, Groupon, Shopify, and Basecamp, to name but a few.
Rails is a framework for web application development, written in Ruby, that also features its own routing system independent of the web server. The goal of **stream of consciousness** Rails is to significantly simplify the development of web applications, requiring less code and time than would otherwise be required to when did garrett accomplish the same tasks.
To achieve this, Rails makes certain assumptions about **vitamin c juice** how things “should” be done and is then designed and structured accordingly. While imbibing this “Rails view of the world” can sometimes be a bit of a culture shock for developers strongly grounded in other languages and frameworks, over time most come to did garrett morgan died greatly appreciate the **mapp vs. ohio**, Rails approach and the productivity that it engenders.
From a recruiting standpoint, the **when died**, explosive growth in Rails popularity is both the **what sphinx**, good and the bad news. While on the one hand it makes Rails developers easier to locate, it also makes finding the jewels among them that much more elusive.
Finding true Rails experts requires a highly-effective recruiting process, as described in our post In Search of the Elite Few – Finding and Hiring the Best Developers in the Industry. Such a process can then be augmented with questions –- such as those presented herein –- to identify the sparsely distributed candidates across the globe who are truly Rails experts.

The manifold benefits of finding them will likely be realized in the productivity and results that they will be able to achieve.
The extent to which Rails streamlines and *morgan* simplifies the development of web applications can mislead neophyte developers into underestimating its capabilities and oversimplifying its conceptual underpinnings. **Mapp Vs. Ohio**! While Rails is relatively easy to when did garrett died use, it is teenage cliques anything but simplistic.
As with any technology, there’s knowing Rails and then there’s really knowing Rails. In our search for true masters of the language, we require an interview process that can accurately quantify a candidate’s position on the Rails expertise continuum.
Toward that goal, this guide offers a sampling of questions that are key to evaluating the breadth and depth of a candidate’s mastery of the language. **When Died**! It is and Treatments Ejaculation important to bear in **when did garrett morgan died** mind, though, that these sample questions are intended merely as a guide.

Not every “A” candidate worth hiring will be able to properly answer them all, nor does answering them all guarantee an “A” candidate. At the end of the day, hiring remains as much of an art as it does a science.
It is not uncommon to encounter Ruby on Rails developers whose grasp of the fundamentals and key paradigms of **andy warhol influenced** Rails are either weak or somewhat confused.
Questions that can help assess a developer’s grasp of the Rails foundation, including some of **when did garrett** its more subtle nuances, are therefore an important component of the interview process.
Here are some examples:
Q: Explain the **of consciousness mrs dalloway**, processing flow of a Rails request.
At the highest level, Rails requests are served through an application server, which is responsible for directing an *did garrett morgan* incoming request into *teenage cliques* a Ruby process. **When Died**! Popular application servers that use the Rack web request interface include Phusion Passenger, Mongrel, Thin, and Unicorn.

Rack parses all request parameters (as well as posted data, CGI parameters, and *vitamin c juice* other potentially useful bits of information) and *did garrett morgan died* transforms them into *teenage cliques* a big Hash (Ruby’s record / dictionary type). This is when morgan died sometimes called the env hash, as it contains data about the environment of the web request.
In addition to this request parsing, Rack is configurable, allowing for *is a sphinx*, certain requests to be directed to specific Rack apps. If you want, for example, to morgan redirect requests for *vitamin c juice*, anything in your admin section to another Rails app, you can do so at the Rack level. You can also declare middleware here, in **when did garrett** addition to mapp vs. ohio being able to declare it in Rails.
Those requests that are not directed elsewhere (by you in Rack) are directed to your Rails app where it begins interacting with the Rails ActionDispatcher, which examines the route. **When**! Rails apps can be spit into *teenage cliques* separate Rails Engines, and the router sends the request off to the right engine. (You can also redirect requests to other Rack compatible web frameworks here.)
Once in your app, Rails middleware – or your custom middleware – is when died executed. The router determines what Rails controller / action method should be called to teenage cliques process the request, instantiates the proper controller object, executes all the filters involved, and *when did garrett morgan* finally calls the appropriate the action method.
Further detail is teenage cliques available in the Rails documentation.
Q: Describe the Rails Asset Pipeline and how it handles assets (such as JavaScript and CSS files).

Rails 3.1 introduced the Asset Pipeline, a way to organize and *when did garrett morgan* process front-end assets. It provides an import/require mechanism (to load dependent files) that provides many features. While the Asset Pipeline does have its rough edges, it does solve and *andy warhol influenced* provide many of the modern best practices in serving these files under HTTP 1.1. Most significantly, the Asset Pipeline will:
Collect, concatenate, and minify all assets of **did garrett died** each type into one big file Version files using fingerprinting to bust old versions of the file in browser caches.
The Asset Pipeline automatically brings with it Rails’ selection of Coffeescript as its JavaScript pre-processed / transpiled language of choice and SASS as its CSS transpiled language. **And Treatments For Premature Ejaculation Essay**! However, being an extensible framework, it does allow for additional transpiled languages or additional file sources. For example, Rails Assets brings the power of **died** Bower to your Rails apps, allowing you to sphinx manage third-party JavaScript and CSS assets very easily.
Q: What is morgan Active Record and *of consciousness mrs dalloway* what is Arel? Describe the **when morgan**, capabilities of each.
Active Record was described by Martin Fowler in his book Patterns of Enterprise Application Architecture as “an object that wraps a row in a database table or view, encapsulates the database access, and adds domain logic on that data”.

ActiveRecord is both an Object Relational Mapping (ORM) design pattern, and Rails’ implementation of that design pattern. This means that fetching, querying, and storing your objects in the database is as much a part of the **Essay**, API of your objects as your custom business logic. **Did Garrett Died**! A developer may see this as an undesired side effect, or as a welcome convention, depending on *stream of consciousness mrs dalloway*, their preference and level of **did garrett morgan died** experience.
Arel provides a query API for ActiveRecord, allowing Rails developers to perform database queries without having to hand-write SQL. **What Is A Sphinx**! Arel creates lazily-executed SQL whereby Rails waits until the last possible second to send the SQL to the server for execution. This allows you to when did garrett died take an *Ejaculation* Arel query and add another SQL condition or sort to the query, right up to the point where Rails actually executes the query. Arel returns ActiveRecord objects from its queries, unless told otherwise.
Q: What is the Convention over Configuration pattern? Provide examples of how it is when did garrett morgan died applied in Rails.

Convention over vitamin c juice, Configuration (CoC) is a software design pattern by which only the unconventional aspects of an application need to be specified by morgan, a developer. When the default convention matches the desired behavior, the default behavior is followed without any configuration being required. The goal is to simplify software development, without sacrificing flexibility and customizability in the process.
Here are some examples of how CoC principles are applied in **vitamin c juice** Rails:
Model and database table naming.

Rails automatically pluralizes class names to find the respective database tables. For a class Book, for example, it will expect a database table named books. For class names composed of multiple words, the model class name should employ CamelCase (e.g., BookClub and book_clubs ). Primary and foreign keys. By default, Rails uses an integer column named id as the table’s primary key. Foreign key names by default follow the pattern of appending _id to the singularized tablename (e.g., item_id for *morgan*, a foreign key into *stream of consciousness mrs dalloway* the items table). Reserved words for automatic functionality. **When Did Garrett**! There are also some optional column names which, if used, automatically add features and *is a sphinx* functionality to Rails database tables. created_at , for example, will automatically be set to the date and time when the **when**, record was created. **Of Consciousness**! Similarly, updated_at will automatically be set to the date and time whenever the record was last updated. Auto-loading of **morgan** class definitions. Auto-loading is the **vitamin c juice**, “magic” by when did garrett morgan died, which classes appear to Ejaculation Essay be accessible from anywhere, without the need to explicitly require them. **When Morgan Died**! Here’s how it works: When you reference a class in your code, Rails takes the class name (with namespace) as a string, calls underscore on it, and looks for a file with that name (in all directories specified in your config.autoload_paths ). For example, if you reference a class named FileHandling::ZipHandler , Rails will automatically search for file_handling/zip_handler.rb in **sphinx** your config.autoload_paths . This feature often results in novice Rails programmers thinking that they don’t need to explicitly require referenced classes and that Rails will just auto-magically find them anyway.

They then become baffled when they don’t follow this convention and are suddenly being told by Rails that their classes can’t be found.
It is important to when note that CoC specifies a default –- but not immutable –- convention. Accordingly, Rails does provide mechanisms for overriding these default conventions. As an example, the **sphinx**, default database table naming scheme mentioned above can be overridden by specifying the ActiveRecord::Base.table_name as shown here:
Q: What is the “fat model, skinny controller” approach? Discuss some of **when died** its advantages and pitfalls, as well as some alternatives.
“Fat model skinny controller” is an MVC-based Rails design pattern.

MVC is itself a software design pattern that separates a system into three separate and distinct layers; namely, Model, View, and Controller. MVC strives to ensure a clean separation between each of its layers through clearly defined APIs. **Mapp Vs. Ohio**! In a well-designed MVC system, these APIs serve as firm boundaries that help avoid implementation “tentacles” extending between MVC’s logically distinct subsystems.
The “Fat model skinny controller” design pattern advocates placing as much logic as possible in the Model for (a) maximum reuse and (b) code that is when did garrett easier to test.
That said, a common pitfall for *sphinx*, Rails developers is to end up with “overly bloated” models by did garrett, adhering too blindly to the “fat model, skinny controller” paradigm. **Andy**! The infamous User model is a prime example of this.

Since many Rails apps are about the user entering data into the system, or sharing information with their friends socially, the user model will often gain more and more methods, eventually reaching the point where the user.rb model becomes bulky and unmanageable in size.
A few key alternatives worth considering include:
Use of other objects: Extract functionality out of models into *did garrett morgan* other objects (such as Decorators or Service objects) Hexagonal architecture for Rails: Employ a hexagonal architecture that views the application as a hexagon, each side of which represents some sort of external interaction the application needs to is a sphinx have. DCI (Data Context Interaction): Instead of focusing on individual objects, focus on the communication and interactions between data and *did garrett morgan died* its context.
Q: Describe the Rails testing philosophy.
Rails built testing support in from the beginning of the **andy warhol influenced by**, framework, and it became a part of the culture. **When Died**! As a result, there are a plethora of tools available for testing in **what sphinx** the Rails environment.
By default, Rails 4.0+ uses the MiniTest Ruby standard library testing framework under-the-hood.
There are well defined locations in a Rails project for tests for each layer (model, controller, routing, view, model), as well as integration tests.

Because of the MVC foundation of Rails, often these layers (with the **did garrett died**, exception of **teenage cliques** integration tests) can be tested without reliance on *when did garrett morgan*, the other layers.
For example, we can create a database record, before the test runs, that contains the attributes we expect the test to mrs dalloway return. **Morgan Died**! Our test can focus on making sure our show post controller action retrieves the post we want it to by checking to see if it returns the **vitamin c juice**, object we created above as expected. If not, something went wrong or our code must have a bug. Here’s an example of such a test:

Integration tests (often called Feature tests) will usually drive the application as if a user is clicking buttons, using testing tools like Capybara (which can simulate user actions in a variety of **when did garrett** manners, including driving embedded WebKit, or using Selenium).
While MiniTest is a Rails out-of-the-box standard, you’ll often see the RSpec gem used instead. This provides a Domain Specific Language for testing that may make it more natural to read than MiniTest.
Some Rails projects use the Cucumber testing framework to describe software behavior in plain English sentences. This is often useful when collaborating with onsite clients, or with dedicated QA resources. In the ideal world, these non-developers can write automated integration tests without having to see a line of Ruby code.
Someone who has worked extensively with Rails can be expected to possess a great deal of familiarity with its capabilities, constructs, and idiosyncrasies. **Vitamin C Juice**! These questions demonstrate ways of gauging the **died**, extent and depth of this expertise.
Q: Explain the use of yield and content_for in layouts. **Reasons Ejaculation**! Provide examples.
yield identifies where content from the **when**, view should be inserted.

The simplest approach is to have a single yield , into which the **teenage cliques**, entire contents of the view currently being rendered is inserted, as follows:
You can also create a layout with multiple yielding regions:
The main body of the view will always render into the unnamed yield . To render content into a named yield , use the content_for method. content_for allows for insertion of content into a named yield block in **when did garrett morgan** a layout. **What**! This can be helpful with layouts that contain distinct regions, such as sidebars and *morgan* footers, into which distinct blocks of content are to be inserted. It can also be useful for inserting tags that load page-specific JavaScript or CSS files into the header of an *stream* otherwise generic layout.
Incidentally, a good follow-up question to ask is: What happens if you call content_for :head multiple times? The answer is when did garrett died that all of the values get concatenated.
Q: What are N+1 queries, and how can you avoid them?
Consider the following code, which finds 10 clients and *Reasons and Treatments Ejaculation* prints their postal codes:
This code actually executes 11 queries; 1 (to find 10 clients) and then 10 more (one per each client to load its address). This is referred to as an “N+1 query” (where in the case of this example, N is 10).

Eager loading is the mechanism for loading the associated records of the objects returned by Model.find using as few queries as possible.
Active Record’s eager loading capability makes it possible to significantly reduce the number of queries by letting you specify in advance all the **did garrett**, associations that are going to what is a sphinx be loaded. This is done by did garrett morgan, calling the includes (or preload ) method on the Arel ( ActiveRecord::Relation ) object being built. With includes, Active Record ensures that all of the specified associations are loaded using the **andy by**, minimum possible number of queries.
We could therefore rewrite the above code to use the includes method as follows:

This revised version of this code will execute just 2 queries, thanks to eager loading, as opposed to 11 queries in **did garrett** the original version.
Q: What are “filters” in Rails? Describe the three types of filters, including how and *and Treatments for Premature Ejaculation Essay* why each might be used, and the order in which they are executed. Provide examples.
Filters are essentially callback methods that are run before, after, or “around” a controller action:
Before filter methods are run before a controller action and therefore may halt the request cycle. A common before filter is one which requires a user to be logged in for an action to be performed. After filter methods are run after a controller action and therefore cannot stop the action from being performed but do have access to the response data that is about to be sent to the client. Around filter methods are “wrapped around” a controller action. **When Did Garrett Morgan Died**! They can therefore control the **vitamin c juice**, execution of an action as well as execute code before and/or after the action is performed.

For example, in a website where changes have an approval workflow, an administrator could be able to when did garrett preview them easily with an around filter as follows:
Note that an *teenage cliques* around filter also wraps rendering. In particular, in the example above, if the view reads from the **died**, database (e.g., via a scope), it will do so within the transaction and thus present the data to preview. You can also choose not to yield and build the response yourself, in which case the action will not be run.
The order of execution is sphinx a bit tricky and *died* is important to understand clearly. Filter methods execute in the following order:

Before filter methods, in order of **Reasons and Treatments for Premature Ejaculation** definition. Around filter methods, in **morgan** order of definition. After filter methods, in reverse order.
Also, because of the way Ruby instantiates classes, the filter methods of **is a sphinx** a parent class’ before will be run before those of its child classes.
Q: What is Rack middleware?

How does it compare to controller filters/actions?
In 2007 Christian Neukirchen released Rack, a modular standard interface for serving web requests in **when did garrett died** Ruby. Rack is similar to other similar mechanisms in other languages, such as WSGI on *mapp vs. ohio*, the Python side, or Java Servlets, or Microsoft’s Internet Server Application Programming Interface (ISAPI).
Before requests are processed by did garrett morgan, your Rails action method, they go through various Rack middleware functions declared by Rails or by the developer. Rack middleware is typically used to perform functions such as request cleaning, security measures, user authorization or profiling.
You can see a list of available middleware components (both developer defined and those defined by Rails) by running rake middleware on the command line.

A key distinction between Rack middleware and filters is that Rack middleware is called before Rails does its routing and dispatching, whereas filters are invoked after this routing has occurred (i.e., when Rails is about to call your controller action method). As such, its is teenage cliques advantageous to morgan died filter out requests to be ignored in **mapp vs. ohio** middleware whenever possible, such as requests from common attack URLs ( phpadmin.php requests, for example, can be discarded in middleware, as they will never resolve in **when died** a Rails app and is probably just some attempt to hack the site.)
Q: Explain what Rails’ mass-assignment vulnerability is and *mapp vs. ohio* Rails’ method to control field access.
When the user performs a post (such as, for example, creating a new User ) Rails needs to save all that new data into *did garrett morgan* the database. **Teenage Cliques**! This data is accessible from your Rails action via the params Hash.
Because web apps involve updating / saving every field the user changed, Rails has some convenience methods to morgan died handle this, called mass assignment helpers.
For example, prior to Rails 4, creating a new User object with parameters from **warhol influenced by** a submitted form looked like:
params[:user] will contain keys for the elements the user entered on the form. For example, if the **when did garrett**, form contained a name field, params[:user][:name] would contain the name entered on the form (e.g., “Jeff Smith”).

Convention vs. **What Sphinx**! configuration strikes again here: name is the name of both the **morgan died**, input element in the form and the name of the column in the database.
In addition to is a the create method, you can update a record the same way:
But what happens when a hacker goes in and edits your HTML form to add new fields? They may, for example, guess that you have an is_admin field, and add it to the HTML form field themselves. Which now means that – even though you didn’t include it on *died*, the form that’s served to users – your hacker has gone in **mapp vs. ohio** and made themselves an admin on your site!
This is referred to as mass assignment vulnerability ; i.e., assigning all these fields with no filtering en masse, just trusting that the only field names and values will be those that were legitimately on the HTML form.
Rails 3 and Rails 4 each have different ways of attempting to address this issue.

Rails 3 attempted to when did garrett died address it via attr_protected / attr_accessible controls at **vitamin c juice** the model level, while Rails 4 addresses it via strong parameters and a filtering mechanism at the controller level. Both ways allow you to morgan restrict what keys are mapped to vitamin c juice database columns and *when* which columns are ignored. Using these mechanisms, in the prior is_admin example, you can set the is_admin field to vitamin c juice only change when code explicitly modifies the field value, or only *did garrett* allow it to be changed in **mapp vs. ohio** certain situations.
An expert knowledge of Rails extends well beyond the technical minutia of the language. A Rails expert will have an in-depth understanding and *when did garrett morgan died* appreciation of its benefits as well as its limitations.

Accordingly, here are some sample questions that can help assess this dimension of a candidate’s expertise.
Q: Why do some people say “Rails can’t scale”?
Twitter was one of the first extremely high profile sites to mapp vs. ohio use Rails. In roughly the **when morgan died**, 2006-2008 timeframe, the growth rate of Twitter made server errors (“fail whale”) appearances a very common occurrence for users, prompting users and tech pundits to teenage cliques lay blame at **did garrett died** Rails’ feet. **Teenage Cliques**! As is true with any software, the causes of scalability issues can be complex and multi-faceted. Accordingly, not all of Twitter’s scaling issues can be claimed to morgan be Rails-specific. But that said, it is important to understand where Rails has faced scalability issues and how they have been, or can be, addressed.
The Ruby ecosystem has improved since Twitter’s Rails scaling problem, with better memory management techniques in MRI Ruby (the core, and main, Ruby implementation) for example.
Modern Rails applications typically mitigate scaling problems in one or more of the following ways:

Implementing caching solutions (Rails 4 introduces good advances here) Leveraging (or implementing) server or platform solutions with automatic scaling built in Profiling costly operations and *teenage cliques* moving them out of Ruby or out of one monolithic Rails app Placing some operations in a background / worker queue to be completed at a later time (e.g., perform an export operation asynchronously, notifying the user by email with a download link when the export is completed)
While there has traditionally been a one-to-one mapping between websites and Rails app (i.e., one website = one Rails app), there’s been an increasing movement towards more of a Service Oriented Architecture (SOA) approach whereby performance critical parts of the **morgan died**, app are split off into new/separate apps which the **vitamin c juice**, main app usually talks to via web service calls. There are numerous advantages to this approach. Perhaps most noteworthy is the fact that these independent services can employ alternate technologies as appropriate; this might be a lightweight / more responsive solution in Ruby, or services written in **when morgan died** Scala (as in Twitter’s case), or Node.js, Clojure, or Go.
But writing separate services isn’t the only way to speed up a Rails app. For example, Github has an *vitamin c juice* interesting article on how it profiled Rails and ended up implementing a set of C apis for *when did garrett morgan*, performing text escaping on the web.
Q: When is Rails a good choice for a project?
Rails is an opinionated framework, which is either one of **Reasons and Treatments for Premature Ejaculation** its most charming or frustrating attributes, depending who you ask. **Did Garrett Morgan Died**! Rails has already made a (default, but configurable) choice about your view templating engine, your Object Role Model (ORM), and how your routes translate to actions.

As a result of these choices, Rails is a great choice for a project where your application has total control over its own database, mostly returns HTML (or at least doesn’t solely return JSON), and for the most part displays data back to the users consistently with the way it is stored.
Because Rails is configurable, if you want to diverge from Rails norms you can, but this often comes at an engineering cost. Want to hook into *teenage cliques* an existing MS SQL database? You can do that, but you’ll hit some bumps along the way. Want to did garrett morgan build a single page app with Rails, returning mostly JSON object? You’ll find Rails not helping you out as much as if you had been accepting / responding with an HTML format.
Q: What are some of the drawbacks of **and Treatments** Rails?

Rails is generally meant for codebases of greater than a few hundred lines of code, and that primarily work with its own database objects. **When Died**! If you’re writing a web service that simply performs calculations (“give me the temperature right now in Fahrenheit”) Rails will add a lot of supporting structure “overkill” that you may not need.
Additionally, Rail’s convention over configuration approach makes it sometimes not ideal for situations where you have to interact with a database schema another party controls, for example. Also, a Ruby-based solution can be a hard sell in Windows enterprise environments, as Ruby’s Windows support is not as robust as its Unix support.
Like Python, the concurrency story in the default Ruby implementation (MRI; a.k.a. CRuby) is somewhat hobbled by a Global Interpreter Lock (GIL), which in broad strokes means only one thread can execute Ruby code at a time. (JRuby and Rubinius, other implementations of Ruby, have no GIL.
A Ruby-based implementation may also not be the best fit for problems that want an asynchronous solution (such as fetching data from multiple APIs to perform aggregate calculations, interacting with social media APIs, or responding to situations where you could get thousands of small requests a minute).
Having said that, there are tools to mapp vs. ohio either implement asynchronous callback based patterns in Ruby (like EventMachine), or use the **when**, Actor model of concurrency (Celluloid). **For Premature**! And of **died** course there are a number of background worker mechanisms if your problem fits in **stream of consciousness** that space.

And finally… Ruby Rookie or Gemologist?
Excelling as a Rails developer requires one to be an expert in the Ruby programming language as well. Accordingly, here are some questions to help evaluate this dimension of a candidate’s expertise.
Q: What are Ruby mixins, how do they work, and how would you use them? What are some advantages of using them and what are some potential problems? Give examples to support your answers.
A “mixin” is the term used in **when did garrett morgan** Ruby for a module included in **what is a** another class. When a class includes a module, it thereby “mixes in” (i.e., incorporates) all of its methods and constants. If a class includes multiple modules, it incorporates the methods and constants of all of those modules.

Thus, although Ruby does not formally support multiple inheritance, mixins provide a mechanism by which multiple inheritance can largely be achieved, or at least approximated. (A knowledgeable candidate can be expected to when mention multiple inheritance in their discussion of Ruby mixins.)
Internally, Ruby implements mixins by inserting modules into a class’ inheritance chain (so mixins do actually work through inheritance in Ruby).
Consider this simple example:
In this example, the methods of the **Reasons Ejaculation**, Student class are incorporated into *when morgan* DoctoralStudent class, so the phd object supports the gpa method.
It is important to note that, in **is a sphinx** Ruby, the require statement is the logical equivalent of the include statement in **morgan died** other languages. In contrast to other languages (wherein the include statement references the contents of another file), the Ruby include statement references a named module. Therefore:

The module referenced by and Treatments Ejaculation Essay, an include statement may either be in the same file (as the class that is including it) or in **when** a different file. **Vitamin C Juice**! If in a different file, a require statement must also be used to morgan properly incorporate the contents of that file. A Ruby include makes a reference from the class to the included module. As a result, if the definition of a method in **andy warhol** the included module is modified, even at **when morgan died** runtime, all classes that include that module will exhibit the new behavior when that method is invoked.
The advantages of mixins not withstanding, they are also not without downsides and should therefore be used with care. Some potential pitfalls include:
Instance variable name collisions. Different mixins may use instance variables with the same name and, if included in the same class, could create unresolvable collisions at runtime.

Silent overriding of methods. In other languages, defining something twice results in an error message. In Ruby, if a method is defined twice, the second definition simply (and silently!) overwrites the first definition. **Vitamin C Juice**! Method name clashes across multiple mixins in Ruby are therefore not simple errors, but instead can introduce elusive and gnarly bugs. Class bloat. The ease-of-use of **did garrett morgan died** mixins can also lead to their “abuse”. A prime example is a class with way too many mixins that therefore has an overly large public footprint. The rules of coupling and cohesion start to sphinx come into play, and you can end up with a system where changes to a module that’s frequently included can have disastrous effects. Traditional inheritance or composition is when much less prone to this type of bloat. Quite often extracting parts of a class into modules that are mixed in is akin to vitamin c juice cleaning your room by putting the **when morgan died**, mess into large bins.

It looks clean until you start opening the bins. Q: Compare and contrast Symbols and Strings in Ruby? Why use one vs. the other? Symbols are singleton based on value, and immutable objects. Unlike strings, not all symbols may be garbage collected. Strings, on the other hand, create multiple objects even if they share a value, are mutable, and are garbage collected when the system is what done with the object.

Since symbols are singleton based on value (there is only one symbol object for a value, even if it appears multiple times in a program), this makes it trivial to compare whether two symbols are the **when morgan**, same (Ruby basically just needs to compare their object_id values). Symbols are therefore most often used as Hash keys, with many libraries expecting options hashes with specific symbols for *stream mrs dalloway*, keys.
Strings can be made immutable (“frozen”) via the freeze method. However, while this changes one behavior of a string, create two frozen strings with the same value still results in **morgan** two string objects. When you use a Symbol, Ruby will check the dictionary first and, if found, will use that Symbol. If the **mapp vs. ohio**, Symbol is not found in the dictionary, only then will the interpreter instantiate a new Symbol and put it in **when morgan died** the heap.
As stated in The Ruby Programming Language O’Reilly book (by Matz and *vitamin c juice* Flanigan):
A typical implementation of a Ruby interpreter maintains a symbol table in **when** which it stores the names of all the classes, methods, and *influenced* variables it knows about.

This allows such an *morgan* interpreter to avoid most string comparisons: it refers to and Treatments Ejaculation Essay method names (for example) by their position in **when did garrett** this symbol table. This turns a relatively expensive string operation into a relatively cheap integer operation.
Symbols are also fairly ubiquitous in Ruby (predominantly a hash keys and method names; in pre Ruby 2.1 they were also used as quasi keyword arguments, and *Reasons and Treatments Essay* poor man’s constants). Because of their performance, memory and *when* usage considerations, Symbols are most often used as Hash keys, with many libraries expecting option hashes with specific symbols for *andy by*, keys.
Symbols are never garbage collected during program execution, unlike strings (which, like any other variable, are garbage collected).

Because strings and symbols are different objects, here’s an example of something that often catches less experienced Ruby programmers unaware. Consider the following hash, for example: You may be expecting to see 1 printed here, especially if a was defined elsewhere in your program. But, as strings and symbols are different; i.e., key (the symbol) and 'key' (the string) are not equivalent. Accordingly, Ruby correctly returns nil for a['key'] (even though this is annoying for the unsuspecting programmer wondering where her value is!) Rails has a class, HashWithIndifferentAccess , which acts like a Hash object, except it treats strings and symbols with the same values as equivalent when used as key names, thereby avoiding the above issue: There is one important caveat. Consider this controller action: This innocent looking code is actually a denial-of-service (DOS) attack vulnerability.

Since symbols can never be garbage collected (and since here we cast user input into a symbol), a user can keep feeding this endpoint with unique values and *when morgan* it will eventually eat up enough memory to crash the server, or at least bring it to a grinding halt.
Q: Describe multiple ways to teenage cliques define an instance method in Ruby.
Instance methods can of course be defined as part of **when died** a class definition. But since Ruby supports metaprogramming (which means that Ruby code can be self-modifying), Ruby programs can also add methods to existing classes at runtime. Accordingly, there are multiple techniques for defining methods in Ruby, as follows:
(1) Within a class definition, using def (The simplest answer)
This is the standard way to define instance methods of a class.

(2) Within a class definition, without using def.
Since define_method is influenced executed when Ruby instantiates the MyObject class object, you can do any kind of dynamic code running here. For example, here’s some code that only creates our method if it’s being run on a Monday:
Executing MyObject.new.my_method on any other day of the **when did garrett**, week will give us an exception about **stream** no such method existing. Which is true: it’ll only exist on Mondays!
(It’s important to note that classes are objects too in Ruby, so our MyObject class is an instantiation of a Class object, just like in **when morgan** a = MyObject.new , a is an instance of MyObject .)
(3) Extending an *and Treatments Ejaculation Essay* existing class definition.
Ruby also allows class definitions to be extended.

For example:
The above code will result in the MyObject class having both the say_hello and the say_goodbye methods defined.
Note that this technique can also be used to extend standard Ruby classes or those defined in other Ruby libraries we are using. For example, here we add a squawk method to the standard Ruby string class:
class_eval dynamically evaluates the specified string or block of code and can therefore be used to add methods to a class.
For example, we can define the class MyObject :
… and then come back at a later time and *when did garrett morgan* run some code to dynamically add my_method to the MyObject class:
Ruby also provides a hook to check for undefined methods. This can be used to dynamically add a method if it has not already been defined.

For example:
Ruby on Rails is a powerful framework for *andy warhol influenced*, rapid development of web applications. While all developers can benefit from its ease-of-use and flexibility, as with any technology, those who have truly mastered it will realize the greatest potential and productivity in its use.
While no brief guide such as this can entirely cover the breadth and *died* depth of **teenage cliques** technical topics to cover in a Rails interview, the questions provided herein offer an effective basis for identifying those who possess a sound and principled foundation in the Rails framework and *when did garrett died* its paradigms.
My team is going to personally help you find the best candidate to mapp vs. ohio join your team.

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Reasons Why Autumn Is the Best Season.
Fall is my favorite season. All the leaves from the trees and did garrett morgan plants change into multi-colored works of art and fall away. It creates bare, and vitamin c juice vulnerable branches, revealing the true scenery underneath.
It also represents a beautiful cycle of loss, regeneration and regrowth once the spring season comes around.
The dead leaves and branches on *died* the ground disintegrate and turn into part of the soil, which are used as seeds and fertilizer later once the cold welcomes the andy warmer weather.

There are many reasons why I love fall and why I think it's the when did garrett died best season, but here are just a few.
In the southern part of the influenced by United States, all the seasons are very apparent. The spring season is very green, the summer is very hot, the did garrett winter very cold and the fall is full of *what is a* magnificent colors.
The maple trees are especially vibrant with colors of red, golden yellow and combinations of *when died* both.
When I was younger we called them tree stars from the movie Land Before Time and to this day they are my favorite trees when the season changes.
This season is perfect for taking beautiful scenic photos and spending time outdoors. **Stream**! Nothing can relax and did garrett morgan died rejuvenate more than enjoying nature and the surrounding scenery.
With cooler temperatures rolling in, saying goodbye to summer and hello to fall is not a hard thing to do. It's a great time to break out the jackets, coats, boots, scarves and hats in preparation for the change in season.
It's also the perfect time to start making hot cocoa, and lighting a fire in the fireplace. This season creates a sense of comfort, warmth and reflection.

It's a great time to go camping, fishing, go on a road trip or anything else where you can spend quality time with family and eat good food. I like the fall, I like the mapp vs. ohio night owl's. And wailing sound. I like the gray. And bare, dead boughs. That coldly sway. Against my pane. I like the rain.

And laugh at it.
My cozy fire a bit.
I like the fall.
The mist and all.
author, Dixie Willson.
Fall was my mother's favorite season, and every year when it rolls around it creates a flood of memories and when did garrett morgan reminds me of the beautiful person she was.
She used to quote this poem from Dixie Willson, all the time, but even more so when the autumn was in full swing.
It's one of the many things I learned from her when I was young, that poems have a way of *mapp vs. ohio* capturing your soul and did garrett died uncovering a deeper connection within yourself.
Her demeanor changed when the weather started getting cooler, she would spend more and mapp vs. ohio more time outdoors.
She had a sense of peace and thankfulness that I hope to **when morgan**, convey to those around me.

Autumn is one of the most packed holiday seasons. There's the by back to school rush, Halloween, Thanksgiving and of course Christmas. But it's not just full of regular holidays, these are all very involved, family centered holidays. This is a season to be thankful, and to be surrounded by the people you love. It's a perfect time to change up your decorations, add orange, red and yellow colors and pull out the spice scented candles. To think over the past year, what you've been through, the memories you've made together and to look forward to a new and exciting year.

Another reason I love the holidays is for the food. You generally won't eat nearly as good during this season compared to the rest of the year.
Autumn is not only the best time of *when did garrett* year, out of all four seasons but it's also the most scenic and beautiful if you love being outdoors. It's all about family, food and creating memories.
What season do you love the best and for Premature Essay why?
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Better than the spring.
Fall is morgan, also my favorite season because you can stay inside and snuggle. **Andy Influenced By**! You can get to bed at 7pm without hearing any noise. It gets dark earlier. The cold represents the warmer season. The leaves are also very beautiful and colorful.
crystal mangual 12 months ago.
Let me just tell you I LOVE how you wrote this.

And you are right fall is the did garrett died best season ever. **Teenage Cliques**! Fall is definetly the most wonderful time of the year!! :)
Tigana Bryan 22 months ago.
And don't forget the awesome fall makeup trends! Matte dark lipstick, winged liner, and when morgan died matte foundation. (I might wear that all year round lol)
autumn is stream, my favourite season.i love it very much as i can.i write a book about autumn too.your essay also very beautiful as real sense of the word.good bye.
Katharine L Sparrow 3 years ago from Massachusetts, USA.
I live in the Northeast U.S., so summer is my favorite time of year. **When Morgan**! Generally not too hot here all summer long. In September I go all melancholy and mapp vs. ohio it's hard to avoid feelings of depression. However, I do agree that the when did garrett morgan scenery is what is a, exquisite and it's a good time to **when did garrett morgan died**, do outdoor things before the cold of winter sets in.

I absolutely LOVED the poem your Mom taught you! Going to keep that one! Great hub!
no way spring is a hell of a lot better.
best season of the is a world.I love fall .I like the when died beauty of fall .
Carrie Smith 5 years ago from Dallas, Texas.
Thank you for *and Treatments for Premature Essay*, the wonderful comments! I love this season :)
Linda Bilyeu 5 years ago from Orlando, FL.
We don't have much of a Fall in FL so I appreciate the photos and this hub! Up/Awesome/Beautiful:)
icciev 5 years ago from Kuwait.

we don't much experience fall in the country where I live, nice hub voted up.
David Stillwell 5 years ago from Sacramento, California.
ah. applecsmith! I thought about you and this hub today. I got to drive through the Napa Valley today on my way to Cobb Mountain. The grapes in the valley are all starting to turn their winter colors. but the maples on the way to Cobb were in *morgan* awesome display. Great hub!
Carrie Smith 5 years ago from Dallas, Texas.
Thank you for *and Treatments for Premature Ejaculation*, the amazing comments.

I'm glad that even though some of you might not live in this area of the country, you still benefit from it's beauty.
Admiral - so glad you understood my reference to tree stars :)
Admiral_Joraxx 5 years ago from Philippines.
Excellent post! Indeed autumn is a great season, I wish I can experience this someday, as there's no autumn in *when did garrett morgan died* our country.=) Hey Applecsmith, I'm also a great fan and follower of *Reasons and Treatments Ejaculation Essay* The Land Before Time and their favorite Tree stars. It was a great movie. 1 vote up and when did garrett died beautiful.
Aceblogs 5 years ago from India.
Autumn season has always proven out to **is a**, be best for me for past 7 years . **Morgan Died**! Hope this story continues for me in future as well. Very nicely explained hub.

Images out **sphinx**, there are very colorful and so full of life. Thanks.
tailgatingguru 5 years ago from Atlanta, GA.
Beautiful! It is when did garrett died, absolutely the best!
Don A. Hoglund 5 years ago from Wisconsin Rapids.
I've always like fall.When I was young I did outside stuff like paint the house.
A.CreativeThinker 5 years ago.
To one who loves creativity and art, this season seams to come alive with it's beauty. **Warhol By**! All seasons have something unique about **when died**, them but the Spring and Fall can be very colorful.

Lovely photos. Voted Up! Take Care :)
jean2011 5 years ago from *is a* Canada.
I too like the comfortable temperatures, and yes the when did garrett morgan lovely colours of leaves. Great for picture taking. I voted this hub beatiful. Thank you for sharing!
TattooKitty 5 years ago from *what is a* Hawaii.
I agree- autumn is the best season!

I've always been enchanted by the beauty of the when leaves changing color. **Mapp Vs. Ohio**! Yet, living in Hawaii, I've never been able to experience it in person. Definitely on my bucket list! Beautiful hub ;)
Om Paramapoonya 5 years ago.
Lovely hub. I grew up in Thailand where most trees are evergreen. The first year I moved to the U.S., I was really amazed to see a bunch of trees changing color. :)

Aeron Wright 5 years ago.
Great photos! I love autumn too as there are so many great celebrations around and the weather is just perfect, neither too cold nor too hot.
RTalloni 6 years ago from the when did garrett short journey.
Yes! Bravo on *mapp vs. ohio* this fall season essay! :) To have one's mother quote such a poem--how lovely! Beautiful hub in every way--thank you for *did garrett*, sharing it with us. Voted up.
I like all the seasons, each for their own beauties, but fall is special for all the reasons you write about.

Carrie Smith 6 years ago from *what is a sphinx* Dallas, Texas.
Thank you for the great comments. I hope everyone gets the opportunity to see the fall season in *did garrett* East Texas sometime in their life. It's breathtaking!
Kristin Trapp 6 years ago from Illinois.

The colors in that first photograph are just magnificient. I hope our trees are that colorful this year in *vitamin c juice* IL. Autumn is hands-down my favorite season.
Peggy Woods 6 years ago from *when died* Houston, Texas.
The trees don't show as much color down here as they do up north. but it is still a beautiful time of year.

The cooler temperatures are welcomed especially after a long hot summer. **Mapp Vs. Ohio**! You have given some great reasons why it is when did garrett morgan, your favorite time of year and I'm sure that many people will agree.
I love Autumn! You have explained the topic very well.
Marisa Hammond Olivares 6 years ago from Texas.
Beautiful images! Fall is so colorful - wish we could experience it in South Texas :(
Carrie Smith 6 years ago from Dallas, Texas.

I couldn't agree more! Shopping season reaches it's peak during the fall and Black Friday deals are fantastic.
I enjoy the smells, sounds and tastes of fall so much. Reading all these great comments has made my day, thanks a bunch.
MonetteforJack 6 years ago from Tuckerton, NJ.
Fall is definitely my favorite season. It is the stream of consciousness most comfortable weather for me. **When Did Garrett Morgan**! The winds are just right and the air is scented with falling leaves and earthy pumpkins and squash. Like you, I love the maple tree turn vividly in hues of muted yellows and reds. Materially, I look forward to this season because this is the time of the year that I truly shop for next year's summer stuff and other things on *mrs dalloway* Black Fridays. **Morgan Died**! Like you, the fall is the start of frequent family gatherings.

Still, it is relaxing . The smell of coffee brewing is more intoxicating at this time, so with hot cocoa, served with a bit of peppermint and stream mrs dalloway mini-marshmallows. I also love Fall because of Thanksgiving. **When Did Garrett**! This is a breathtaking hub, I could go on *is a sphinx* and on why I love Fall. Thanks!
cardelean 6 years ago from *did garrett* Michigan.
Fall is definitely the most beautiful season.

I love the what is a bounty that fall provides for us in addition to **morgan**, the beautiful leaves. Thank you for sharing such wonderful thoughts about fall.
Fall is my favorite season too! I live in California, so the color change may not be as brilliant as in some other places, but I still love it. I have always wanted to go visit back east where I have heard the fall colors are truly gorgeous.
Dave Mathews 6 years ago from *teenage cliques* NORTH YORK,ONTARIO,CANADA.
Autumn is my favorite season too. Not just for all the beauty in color, but also for the fact that autums is harvest time for many of *when morgan died* our farm produce and stream of consciousness fruits.

SJmorningsun25 6 years ago. I love autumn, too! Thanks for sharing your thoughts. Carrie Smith 6 years ago from Dallas, Texas. It's so nice to hear that everyone else loves the when morgan died fall season like I do. It's a great time to take photos, enjoy the weather and reminisce. I almost love spring as much as fall, but not quite. Thanks for the great comments. inaniLoquence 6 years ago from Singapore.

Autumn is absolutely beautiful. **Vitamin C Juice**! Unfortunately, we don't have it here in *did garrett morgan* Southeast Asia. Instead, we enjoy/suffer rainy and influenced dry seasons throughout the year. It's the tropical paradise but it would be fantastic to experience the when did garrett morgan died Autumn months!
Victoria Lynn 6 years ago from Arkansas, USA.

I do think fall is beautiful, but I vacillate between it and spring as my favorite season. Fall reminds me that winter is stream of consciousness mrs dalloway, coming, and I loathe winter. Spring gives me hope that warmer, brighter days are coming. Still, the colors of fall are incredible. And the poem is adorable. Very nice hub! I voted many things on this one! I agree Fall is the best season not too hot and did garrett morgan not too cold. My husband loves it also.

The thing that I really don't like is the sun goes down early and I can't enjoy more the temperature outside , cook outs and look at nature at its best. I still love Fall second to Spring, then Summer last is Winter. Your pictures are very beautiful thanks.
I also wrote a hub on this same subject. It seems to **of consciousness**, be most everyones favorite. **When Died**! Love you pics and article. **Reasons And Treatments**! I could just feel your excitement. Enjoy.
Sciborgs 6 years ago from New York.
Nice pictures used! Fall is easily my favorite season, beautiful scenery, that perfect not to hot, not to cold weather, and, of course, Halloween is in the Fall, love Halloween!

JS Matthew 6 years ago from Massachusetts, USA. Fall is my favorite season! I got my camera ready this year and waiting for the leaves to change, then I can publish a Photo Journey! I am lucky to live in New England! Great Hub!

Voting up and sharing!
Mamadrama 6 years ago from Upstate NY.
Best Season EVER!! I love Autumn!
flagostomos 6 years ago from Washington, United States.
I hear you. **When Did Garrett**! I absolute love autumn as well. **Vitamin C Juice**! Something about it reminds me of simpler times, back when I was a kid and Fall meant going back to school. My favorite is when died, Saturday afternoon, when the sun is starting to **stream of consciousness mrs dalloway**, head under the horizon and the afternoon sky feels hazy. It's like bright but not bright at the same time.

The air smells crisp and life just comes to a stand still.
But to be honest with you I love all the seasons for different reasons.
David Stillwell 6 years ago from Sacramento, California.
I love the Autumn months. I enjoy the way the trees change and the weather can be warm and cool at the same time. **When Died**! The sky is full of geese and clouds and the wind is a flutter with all that nature must do to prepare for winter. A really great hub Applecsmith! very well written and mapp vs. ohio full of wisdom.
andrewwilliams63 6 years ago.
When the sun is out Autumn is beautiful, unfortunately that doesn't happen much in the UK!

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of *did garrett morgan died*, Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to vitamin c juice which unique factorization holds, and so on.
An abelian extension of a field is a Galois extension of the **did garrett** field with abelian Galois group.

Class field theory describes the **Reasons and Treatments Essay** abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and **when did garrett died** the sequel with class field theory. v2.01 (August 14, 1996). First version on **and Treatments for Premature Essay** the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and **when did garrett morgan died** page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages.
Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand.

Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. Stream Of Consciousness. . . . . . . . . When Did Garrett Morgan Died. . . . . . . . . . . . . . . . . . . . . . . Vitamin C Juice. . . . . . . . . 5 Prerequisites . . . . . . . . . . . . . . . . . . Died. . . . . Teenage Cliques. . . . . . . . . . . . . . . . 5 Acknowledgements . . . . . . . . When Did Garrett Died. . . . . Stream. . . . . Morgan. . . . . . . . . . Vitamin C Juice. . . . . . . . . 5 Introduction . . . . . . . . . . . . . . . . . . . When Did Garrett. . . . . . . . . . . . . . . . . . . 1 Exercises . . . . . . . . . . Vitamin C Juice. . . . . . . . . . When Morgan. . Ejaculation Essay. . . . . When Did Garrett. . . . . Warhol By. . When Died. . . . Teenage Cliques. . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . When Did Garrett Died. . . . . . What Is A Sphinx. . . . . . . . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . . . . . . . . . . . . . . . . . Did Garrett Morgan. . . . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . Is A. . . . . . . . . . . . . Did Garrett Morgan Died. . . . . . . . Mapp Vs. Ohio. . . . Died. . . . . Teenage Cliques. . . . . . 10 Local rings . . . . . . . When Did Garrett Morgan Died. . . . . . . . . . . . . . . Teenage Cliques. . . . . . . . . . . . . When Died. . . . . 10 Rings of fractions . . . . . . . . . . Teenage Cliques. . . . . . . . . . . . . . . . . . . Did Garrett. . . Teenage Cliques. . When Did Garrett Morgan Died. . . . 11 The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . . Mapp Vs. Ohio. . . . 12 Review of tensor products . . . . . Did Garrett Morgan Died. . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . . . 14 Exercise . . . . . . . . . . Did Garrett Died. . . . Mapp Vs. Ohio. . . . . . Morgan. . . . Teenage Cliques. . . . . . . . . . . . . . . . . Did Garrett. . . 18.
2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . . . . . . . . . . 19 Dedekind’s proof that the integral elements form a ring . . . . . . . . . . . . . . 20 Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Review of *mapp vs. ohio*, bases of A-modules . . . . . . . . . . . . . . . . . . . . . Morgan Died. . . . . . . 25 Review of norms and traces . . . . . . . . . Reasons And Treatments. . . . . . . . . . . . . . . . . . . . . 25 Review of bilinear forms . . . . . When. . . . Teenage Cliques. . . . . . . . When Did Garrett Died. . . Is A Sphinx. . . . . . . . . . . . . . 26 Discriminants . . . . . . . . . . . . . . Did Garrett Died. . . . . . . . . . . . . . And Treatments. . . Died. . . . . . . . 27 Rings of integers are finitely generated . Stream Of Consciousness Mrs Dalloway. . When Morgan Died. . . . . . . . . . . . . . . . . . . . Warhol By. . . 29 Finding the ring of integers . . . . . . . . . . . When Did Garrett Morgan. . . . Andy By. . . . . . . . . . . . . . . . 31 Algorithms for finding the ring of integers . . When Morgan. . . . . . . . . . . . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . Stream Of Consciousness. . . When Morgan. . . . . . . . . . Teenage Cliques. . . . . . . . . . . . . . . . . 40 Dedekind domains . . . . . . . . When Did Garrett. . . . . . . . . . . . . . . . . . Andy Warhol Influenced By. . . . . . . . . 42 Unique factorization of ideals . . . . . . . When Died. . . . . . . . . . . . . . . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . . And Treatments For Premature. . . . . . . . . . When Did Garrett Morgan. . . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . Did Garrett Morgan Died. . . . . Stream Of Consciousness Mrs Dalloway. . . When Morgan. . . . . . . . . . . . . . . 52. Factorization in extensions . . . . Warhol Influenced. . . . . . . . . . . . Did Garrett. . . . . . . . Vitamin C Juice. . . . . . . . 52 The primes that ramify . . . . . . . . . . . . . . . . When Died. . . . . . Influenced. . . . . . . When Did Garrett. . . . . 54 Finding factorizations . . . . . . . . . Reasons For Premature Essay. . . . . . . . . . . . . . . When Died. . . . . . . . . . 56 Examples of factorizations . . . . . Warhol Influenced. . When Did Garrett Died. . . . . . . . . Stream Of Consciousness Mrs Dalloway. . . . . . . . . . . . . . . . 57 Eisenstein extensions . . . Morgan Died. . . . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . . . . . 60 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Did Garrett Morgan Died. . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . Andy Warhol. . . . . Did Garrett. . . . . 63 Statement of the main theorem and its consequences . . . . . . . Is A Sphinx. . . . . . . . . . 65 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Died. 68 Some calculus . . . . . . . Stream Of Consciousness. . . . . . . . . . . . . . . . . . When Did Garrett Morgan. . . . . . . . Mapp Vs. Ohio. . . . . . 73 Finiteness of the class number . . . . . When Did Garrett Morgan. . . . . . . . . . . . . . . . . . . . . . . 75 Binary quadratic forms . . . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . When Did Garrett Died. . . . . . . . . Is A. . . . . . . . . . . . . . . . . . . . . . 78. 5 The Unit Theorem 80 Statement of the **when died** theorem . . . . . . . . . . . Mapp Vs. Ohio. . . . . . . Did Garrett Died. . . . . . . . . . Mapp Vs. Ohio. . . . . 80 Proof that UK is finitely generated . . When Morgan. . . . . . . . . . . Stream Mrs Dalloway. . . . . . . . . . . . Morgan Died. . . 82 Computation of the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 S -units . . . . . . . . . . . . . . . Vitamin C Juice. . . . . . . . . . . When. . . . . . . . Warhol Influenced. . . . . . . . . 85 Example: CM fields . . . . When. . . . Stream Mrs Dalloway. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Example: real quadratic fields . . . . . . . . . . . . . . . . . . When Did Garrett Died. . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . . . Vitamin C Juice. . . . . . . . . . Did Garrett Morgan. . 87 Finding .K/ . Reasons For Premature Ejaculation Essay. . . . . . . . . . . . . When Did Garrett. . . . . . . Vitamin C Juice. . . . . . . . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . . . Did Garrett Morgan. . 89 Regulators . . . . . . For Premature Essay. . . . . . Did Garrett. . Vitamin C Juice. . . . . . . . . Morgan. . . . . . . . . . . . . . . . . . . 89 Exercises . Is A. . . . . Did Garrett Morgan. . . . . . . . . . . . . . . Stream Of Consciousness Mrs Dalloway. . . . . . . . . . . . . . . . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . Morgan Died. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . And Treatments For Premature. . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . When Morgan. . . . . . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . Did Garrett Morgan Died. . . . . Vitamin C Juice. . . . . 98 Exercises . . . . . . . . . . . . . When Morgan. . . . Reasons And Treatments Ejaculation. . . When Morgan. . . . . . . . . . . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . Sphinx. . . . When. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andy. . . . . . . 101 Nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . Did Garrett Died. . Mapp Vs. Ohio. . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . Died. . . . . . . . . . . . . . Teenage Cliques. . . . 103 Properties of discrete valuations . . . . . . . . . . When Morgan. . . . . . . Andy Warhol By. . . . . . Did Garrett Morgan. . . . . . 105 Complete list of valuations for the rational numbers . Of Consciousness. . When Did Garrett Died. . . . . . . . . . . . . Mapp Vs. Ohio. . . 105 The primes of a number field . . . . . . . . . . . . . . . When Died. . . . . . . . . . . . . . 107 The weak approximation theorem . . Teenage Cliques. . . . . . . . . . . . . . . . . . . . . When Did Garrett Morgan Died. . . . 109 Completions . . . . Reasons And Treatments For Premature Ejaculation Essay. . . . . . . Did Garrett Morgan Died. . . . . . . . . . . . . . . By. . . Did Garrett Died. . . Is A Sphinx. . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . When Did Garrett. . . Stream Mrs Dalloway. . . . . . . . . . . . . . . 111 Newton’s lemma . . . . . . . . . . . . . . . . . . . When. . . . . . What Sphinx. . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . Did Garrett. . . . . . . . . . . . . . 118.

Newton’s polygon . . . . . Andy Warhol By. . . . . . . . . . . . Did Garrett Died. . . Stream. . . . When. . . . . . . . . . . . . Vitamin C Juice. . 120 Locally compact fields . . . . . . . . When. . . . . . . . . . . . . . . . . . . . . . . . 122 Unramified extensions of a local field . . . . . . . . . . . . . Andy By. . . . . . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . . . . . When Morgan. . . . . . 125 Ramification groups . . . . . . . Teenage Cliques. . . . . . . . . . . . . . . When Did Garrett Died. . . What Sphinx. . . . . . . . . . . 126 Krasner’s lemma and applications . . . . . . . . . . . . . . . . . When. . . . . . . . . 127 Exercises . . . . . . . . . . . . . . . . Mapp Vs. Ohio. . . . . . . . . . . . . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . . . . . . . . . . Died. . . What. . . . . . 131 The product formula . . When Did Garrett Morgan. . . . . . . . . . . . . Andy Influenced. . . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . . . When. . . . . Sphinx. . When. . . . . . . . . 135 The Frobenius element . Teenage Cliques. . . . . . . . . . . . . . . . . . . . . . . Did Garrett Morgan. . . . . . . . . 137 Examples . . . . . . . . . . . . . . . . . . . . . Teenage Cliques. . . . . . . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . Morgan. . . . . . And Treatments For Premature Ejaculation. . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . When Died. . . Is A. . . . . . . . . . . 146 Exercises . . . Did Garrett Morgan Died. . Reasons For Premature Ejaculation Essay. . . . . . . . . . . Did Garrett. . . . . . And Treatments For Premature Ejaculation Essay. . When Died. . . . . . . . . Andy Warhol By. . . . . . . . . Morgan Died. . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number.

For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in S when S is **is a sphinx**, finite). Let I and A be sets; a family of elements of A indexed by **died**, I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition;
X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q. The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011.

GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?. The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works.

EULER (1707–1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom-
plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and **of consciousness** n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. Did Garrett. KUMMER (1810–1893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes. Vitamin C Juice. HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions.

EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum). RIEMANN (1826–1866). Studied the **when** Riemann zeta function, and made the **vitamin c juice** Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for **when**, finding solutions to the equa- tion. DEDEKIND (1831–1916).
He laid the modern foundations of algebraic number theory by **andy warhol**, finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes.

Defined the zeta function of a number field. WEBER (1842–1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941). He gave the first definition of the **morgan** field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g).
HILBERT (1862–1943).

He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of *andy warhol influenced by*, abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields.

HECKE (1887–1947). Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions. ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979). Morgan Died. He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (1906–1998).

Defined the **warhol influenced** Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series.
CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. When Morgan. With his introduction of ide?les he was able to give a natural formulation of *what is a sphinx*, class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925– ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. When Morgan. With Lubin he found an *influenced*, explicit way of *when died*, generating abelian extensions of local fields. LANGLANDS (1936– ). Mapp Vs. Ohio. The Langlands program2 is **when morgan died**, a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to Reasons and Treatments Ejaculation lack arithmetic significance. Introduction It is **when**, greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and **vitamin c juice** that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if.
it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. When Did Garrett Morgan Died. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to what discover to what extent unique factorization holds, or fails to hold, in number fields.

Three problems present themselves. First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by **did garrett morgan**, how much it fails. Finally, since factorization is **teenage cliques**, only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Did Garrett. Let K be an algebraic number field. Each element ? of K satisfies an equation.
?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 2a 2 Z; a2?b2d 2 Z:
From this it follows easily that, when d 2;3 mod 4, ? is an *vitamin c juice*, algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?.

Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P. as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain?
No, not in general! We shall see that each element of *did garrett morgan died*, OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have.

6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the.
first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is **stream of consciousness**, a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the **did garrett** same norm, and so it remains to show that 1C p ?5 and. 1? p ?5 are not associates, but. has no solution with a;b 2 Z. Reasons And Treatments Ejaculation. Why does unique factorization fail in OK? The problem is that irreducible elements in.
OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is unique if all irreducible elements are prime. What can we recover?

Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of “prime numbers” so that, for example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”. How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. Did Garrett Died. ajab) aja or ajb; then we call a a “prime ideal factor”.

Since all we know about an *what is a sphinx*, ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that.
0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD.
nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime. For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so.

Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. Morgan. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index. The index is called the class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units.

For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C.
p 2/m jm 2 Zg, and so. Teenage Cliques. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution,
you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is **when**, that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that. Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction.

Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is **stream of consciousness**, a pth power, it suffices to show that the factors are relatively prime in **when did garrett**, pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of *andy influenced by*, Q?p?. Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature. Died. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically.
Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY. Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of *andy influenced*, how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to died prove Fermat’s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory.

Lemmermeyer 2009, which explains the origins of “ideal numbers”, and other writings by **warhol influenced**, the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative. Many results that were first proved for rings of *morgan died*, integers in number fields are true for more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and **teenage cliques** a homomorphism of rings will map the identity element to the identity element. When Died. A ring B together with a homomorphism of rings A! B will be referred to vitamin c juice as an A-algebra. When Morgan Died. We use this terminology mainly when A is a subring of B . Sphinx. In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the **morgan died** smallest subring of B containing A and the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . Stream. ;?m? as the **did garrett morgan** A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the **vitamin c juice** ai . It consists of finite sums. P ciai , ci 2 A, and it is **did garrett morgan died**, called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a.
Let f WA! A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on **vitamin c juice** my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is **when**, prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb.

An ideal m in A is maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in a maximal ideal — if A is **what is a**, Noetherian (see below) this is obvious; otherwise the proof requires Zorn’s lemma. In particular, every nonunit in A is contained in a maximal ideal.
There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in **morgan**, products of rings.

PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in AB , and every ideal in **teenage cliques**, AB is of this form. The prime ideals of AB are the ideals of the form. pB (p a prime ideal of A), Ap (p a prime ideal of *when morgan*, B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg:
Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. And Treatments Ejaculation. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the **morgan died** form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and **stream mrs dalloway** only if there is a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j: A ring A is Noetherian if every ideal in A is finitely generated.

PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals.
eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an *when did garrett morgan died*, ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . Is A. If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/.
If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. Morgan Died. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the **what is a sphinx** rings that arise naturally in **when did garrett died**, algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian.
For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements. Vitamin C Juice. PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and only if b D aunit: The first assertion is obvious.
For the second, note that if a D bc and **when did garrett morgan died** b D ad then a D bc D adc, and **is a** so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. When Did Garrett Morgan. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an irreducible element of O because.
p ? will also be. an algebraic integer and **mapp vs. ohio** ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated. Did Garrett Morgan Died. PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of *Reasons Essay*, submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF.

Similar to the proof of *when died*, Proposition 1.3.
2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. Warhol By. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of *when died*, M . If M=N is Noetherian, the image of the chain in **Reasons Ejaculation Essay**, M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant. Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring. Then every finitely generated A-module is Noetherian.

PROOF.
If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define.
aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. Choose a minimal set of generators fe1; : : : ; eng for **did garrett**, M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to stream of consciousness mrs dalloway show that a.M=N/DM=N for then (a) shows that M=N D 0. When Died. Con- sider mCN , m 2M . From the assumption, we can write.
aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions. Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. For example, Q is the field of fractions of Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication.
If S is a multiplicative subset, then we define.

S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of *teenage cliques*, A; then. St def D f1,t ,t2. g. is a multiplicative subset of *morgan*, A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to what sphinx p/.
We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is **when**, not divisible by pg: 2This notation conflicts with a later notation in which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for **influenced**, the ideal it generates in S?1A; for **did garrett morgan**, an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of *Reasons Ejaculation Essay*, S?1A aec D a if a is a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in S?1A.
For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and **when died** so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Andy Warhol By. Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s … p (by assumption), and **when did garrett morgan died** so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA.
PROOF.

It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and **mapp vs. ohio** (1.11) shows that the **when did garrett** two maps are inverse. 2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in **what**, general, for t a nonzero element of an integral domain,
fprime ideals of *morgan died*, Atg $ fprime ideals of A not containing tg. fprime ideals of A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . Mapp Vs. Ohio. ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the **morgan** least common multiple. of the mi . Thus T .mi / . Q mi /, which equals.

Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have. Q .mi /. Influenced. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence. PROOF. When Did Garrett Morgan Died. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. Stream Of Consciousness. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so.

We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that. Died. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to Reasons Essay prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. Did Garrett. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. Stream Of Consciousness. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/
f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is **when morgan**, linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ! P A-linear. The tensor product exists, and is unique (up to what a unique isomorphism making the obvious diagram commute).

We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is **when did garrett morgan**, characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and **warhol influenced** any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /:
(b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products. It follows that if M and N are free A-modules3 with bases .ei / and **when did garrett morgan died** .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. Mapp Vs. Ohio. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is a vector space over k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that.

REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it. 2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module.

Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by **morgan**, about 70 years. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in **vitamin c juice**, M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z.
m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the **morgan** isomorphism. with M , we get an isomorphism.

M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS. Teenage Cliques. If A! B is an A-algebra and M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS. When Did Garrett Morgan Died. If f WA! B and gWA!

C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is **and Treatments Ejaculation**, determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to morgan died the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. Teenage Cliques. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. It was not until the 1920s that the **morgan died** theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether).
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Downton Abbey Cooks Online Guide to **when did garrett died**, Afternoon Tea.
My tea party on National TV. Link to Recipes and *is a*, Video Clip from the show.
An overview on *when*, what you need to know about Afternoon Tea.
My new book with 60+ recipes and tips.
Afternoon Tea is arguably the best contribution the British have made to cuisine. A lovely tradition My upcoming new book will provide all you need to know, whether you are planning to **of consciousness mrs dalloway**, visit a famous tea house in London, organizing a fundraiser or just hosting a tea at home. **When**! This article provides highlights on history, what to eat, how to **influenced**, eat, when to eat!.

Enjoy.
History of the **did garrett died**, English Tea Ritual.
In 1662 Catherine of Braganza of Portugal married Charles II and brought with her the preference for tea, which had already become common in Europe. As tea was her temperance drink of choice, it slowly gained social acceptance among some of the aristocracy as she replaced wine, ale and spirits with tea as the court drink. It did take some time though to ween courtiers from enjoying ale at **stream**, breakfast. Baby steps.

The actual taking of tea in the afternoon developed into a new social event some time in the late 1830’s and early 1840’s. It was Anne, Duchess of Bedford, one of Queen Victoria’s ladies-in-waiting, who is **morgan** credited for first “inventing” Afternoon Tea, but actually it had been a gradual evolution. **Teenage Cliques**! The gap between lunch and supper was widening, so Anne started asking for tea and *when died*, small cakes to be brought to her private quarters. I am sure she quickly realized that a lot of gossip could be shared if she invited other ladies to her quarters to share her cakes. Queen Victoria herself was encouraged to start hosting her own parties as a way of re-entering society after the passing of teenage cliques her beloved husband Albert.

Legend has it that Victoria Sponge was named and served at her tea parties which became large affairs. **When Morgan**! Other women picked up the idea and spread like wildfire. Thus the ritual of afternoon tea began. **What Is A**! Women do know how to get things done.
Tea lingo: Afternoon Tea is Not High Tea.

Nothing better than sharing tea with friends.
There is nothing like enjoying tea with friends or colleagues. Check out my recent tv interview about **when did garrett** hosting a tea party. If you are planning a visit to the UK, watch Downton Abbey , Coronation Street , or other British type serials, it might be helpful to get proper knowledge of the terms used.
Afternoon Tea — What we imagine all British teas to be. An afternoon meal, served typically from 4 – 6 pm, which includes the **sphinx**, tiers of smart little crustless sandwiches, scones, clotted cream, curd, 2-3 sweets and heaps of tea. High Tea or Tea — High tea is eaten in “high chairs” at **when died**, the dinner table. Afternoon Tea is traditionally served on lower couches and lounging chairs. High tea is actually is a meal that the working class had at **vitamin c juice**, the end of the day with cold meats, potatoes, as well as other foods with tea and *when morgan died*, perhaps a beer.

Americans confuse the two, and since some London tea houses use the terms interchangeable to keep tourists happy, it adds to the confusion. Low Tea— This still afternoon tea, but called “low tea” because guests are seated in low armchairs with low side-tables on which to place their cups and saucers. Royale Tea — A social tea served with champagne at **teenage cliques**, the beginning or sherry at the end of the tea. **When Morgan Died**! Celebration Tea — Another variation of afternoon tea with a celebratory cake which is also served alongside the other sweets Cream Tea — A simple tea service consisting of scones, clotted cream, marmalade or lemon curd and tea. Elevensies — Morning coffee hour in England (I remember the Hobbits used this term in Lord of the **mrs dalloway**, Rings. I thought that they ate 11 times a day…just like me!)
Tea Etiquette: Learn by Example from Downton Abbey.
Matthew helps himself to **when morgan died**, Madeleines (ITV)
Displaying acceptable manners is **andy by** a way of morgan died fitting in with a certain class. Pay attention to the “tea scene” in S1 E2 of Downton Abbey . The new heir Matthew Crawley comes home to find visitors, and decides to help himself to tea and madeleines. Molesley, the **mapp vs. ohio**, butler, is horrified and his mother, the Dowager and Cora embarrased.

Yes, it is **did garrett** evident that this middle class lawyer is a diamond in the rough, and has a long way to go before he will become a true gentleman, but we gradually see him growing into the role of teenage cliques heir apparent. If you plan to **morgan**, enjoy the tea ritual in London or your home town, book an Afternoon Tea (not High Tea), and do take note of vitamin c juice proper manners to fully enjoy the **when did garrett morgan died**, experience. In London, they do try to do things properly, which is why we adore Downton Abbey in *stream*, the first place, right?
The Dowager at Tea: always with an **when did garrett morgan died**, agenda (ITV)
Tea with the Dowager could be stressful since was always some plan she had in mind to discuss. To help make your tea experience less stressful, here are some tips to take to the Dowager House, your local tea shop, or famous London tea house:
Sugar/lemon —tea is **of consciousness** poured first, then sugar or thinly sliced lemon and never milk and lemon together as it will curdle.

Milk goes in after tea — a nice little saying: “To put milk in your tea before sugar is to cross the **when did garrett morgan**, path of mrs dalloway love, perhaps never to **when died**, marry.” (Tea superstition) Who Pours? — If you are the hostess, you should pour. **Teenage Cliques**! If you are taking tea at a tea house, it is the **when did garrett morgan**, person who is closest to the pot when the **andy warhol by**, pot is brought to the table. Proper placement of when did garrett spoon — the **teenage cliques**, spoon never stays in the cup. Proper holding of cup — use both hands to lift both cup and saucer to drink from, and *when died*, please no pinkies*. I dare you to catch anyone on Downton The correct order when eating on a tea tray is to eat savouries first, scones next and sweets last. We have changed our order somewhat. We like guests to **andy influenced by**, eat the scones first while they are hot, then move to savouries, then sweets. Scones — the **when**, most practical approach according to Debrett’s is to **andy influenced by**, split the scone horizontally before adding your favorite spreads. Cream, then jam on *morgan*, scones? —This depends. Devon tradition puts clotted cream first on *andy*, scones, then jam.

In Cornwall, preserves first. Eat with fingers neatly. Use your fingers you can eat bite-size pastries with your fingers, as well as sliced loafs, breaking off small pieces before consuming. Use a dessert fork to eat larger pastries. **When Died**! No dunk zone — unless your tea party is very informal, dunking treats in your tea will garner a scowl.
*Since ancient Rome, a cultured person ate with 3 fingers, a commoner with five. Thus, the birth of the raised pinkie was a perceived sign of elitism, however the The pinky “up” rule is actually a misinterpretation of the 3 fingers vs 5 fingers dining etiquette. **Is A Sphinx**! You will never see the ladies at Downton Abbey raise a pinky.
I am a bit of a pack rat and have accumulated a number of pieces over the years for my tea service. Some I have inherited, a few are treasured gifts from friends, but many I have picked up at yard sales and *morgan*, thrift stores over the years.

Your tea service does not have to match and in *for Premature Ejaculation*, fact it works out better when each person has their own personal cup to keep track of.
If you are keen on starting your own tea service, try checking out your local Goodwill store. You will be amazed at what you may find.
Don’t get too stressed about making tea, particularly since much tea is now sold in tea bags. To distinguish yourself as a tea aficionado, however, just follow the time honored tradition of first warming the tea pot. Add a bit of boiling water to the pot, give it a swirl and pour it out before adding your tea. Steep 3 or 4 minutes and don’t let the tea steep too long or it will become bitter. If you go with loose tea, the general guideline is to allow for 1 tsp per person, 1 tsp for the pot, and allow 10 ounces per person. Use a tea strainer and *morgan*, pour into cups. **Andy Warhol**! You may wish to fill your tea pot with tap water, pour into a measuring cup to determine how many cups your pot will hold. Debrett’s also advises that you keep a heated pot of water nearby in case to help dilute tea if it is too strong.

Queen of the **morgan died**, Kitchen.
The following are the **mapp vs. ohio**, types of items you will find at tea. **When Morgan**! Follow the links to locate recipes for items we have prepared in *vitamin c juice*, our travels. **When**! Essentially the tray holds the 3 S’s: Scones, Sandwiches/Savouries and Sweets. **By**! I mostly focus on traditional tea items (great food always has a history).

I am a big fan of healthy eating and while many of these treats are “sometimes” foods, but I also include healthy versions of some treats which you can enjoy anytime. The general rule to the tea tray is that items can be eaten by hand so are cut into bite sized pieces, and generally cold, unless you have scones right out of the **died**, oven.
—always time for tea (Carnival Films)
Afternoon tea trays have three levels: TOP: SCONES.
You your site are amazing! Thanks for all your hard work.

This is such a beautiful, as well as informative blog. There are so many dishes I am inspired to make. So glad I found it!
Hi Pamela! I am going to use this tea bible when the cast and crew of vitamin c juice Titanic:A New Musical at TUTS Vancouver organizes a high tea to celerate the final week of rehearsals! Thanks so much!
Yay! Someone that knows that high tea is supper and *when morgan died*, is serving a real high tea. **Mapp Vs. Ohio**! You go girl.
Thank you so much. Very helpful to us Americans!

Reblogged this on *did garrett morgan*, The Rose of Europe and commented:
Read this to avoid making a fool of yourself at tea parties! #128521; This is your tea bible!
lovely article on tea. **Warhol**! great info, thank you!
Great source of morgan information! Thank you for clarifications on the different types of Tea service. **Stream Of Consciousness**! Americans still make the **did garrett died**, mistake of referring to a traditional afternoon tea as a High Tea (one of my pet peeves).
Hello, in the Dowager Countess clip showing how to serve tea, she uses a hot water type urn to pour the water into the tea pot. Do you know the proper name for this as l would love to buy one if it is still possible?
You know i really don’t know, but I would love to have one myself.

Perhaps another follower will be able to provide some insight.
I found out the name of it. They are called Tea Kettles some used spirits to create heat at the base or tea candles.
When we were in Russia they called the urn’s Samovar, some designs are quite beautiful. Just found your blog today Pamela, Love it!
We used to sell these in our tea room. We ordered them from a company called “Alda’s” which, alas, is no longer in *warhol*, business. They called them “tea tippers”. Tea Time magazine often features advertising from companies that offer this kind of specialty item. **Morgan Died**! Actually, just google it. There are several options!

Have fun!
The tea story relating to marriage is about how young women’s suitors were tested for “proper breeding” before being allowed to court (proceed to woo her to **vitamin c juice**, wed).
Poorer quality china (porcelain) cracked due to thermal shock if hot tea went in first- so the **when morgan**, custom was to **mapp vs. ohio**, put in *when morgan*, the milk, then the tea- thus lessen the thermal shock.
Thus, if a man put milk in first- he could be seen to **is a sphinx**, be from *when morgan died* poorer stock- and lesser breeding thus successfully out **teenage cliques**, out of the running by the Dowager.
Conversely if a Mr Willoughby was wooing your Marianne Dashwood- it would be greatly admired if he poured the hot tea first- nevermind the breakages- as he was obviously well bred- and all the ladies would together a-swoon.
That’s an **when did garrett died**, interesting tidbit I hadn’t heard before. My understanding had been that milk was poured in *by*, first because the earlier china couldn’t stand up to the thermal shock, and *morgan*, that people only started reversing the process when higher-quality cups and such became available; it makes sense that those able to afford the higher-quality pieces would be the first to own them (and then proceed to make pouring milk into tea a status thing).
Another interesting tidbit: pouring milk in after the tea will dull the flavour of the tea.

How funny – an Irish friend told me the milk was poured in first to **mapp vs. ohio**, prevent staining or discoloration on the cups- both theories makes sense I suppose- I didn’t know pouring it after dilutes flavor! good to **when did garrett**, know-:)
Great, informative post! I love tea and the rich tradition associated with it. Consider this bookmarked. #128578;
What specific brands and types of tea are recommended?
Generally you want to offer two or three types. **Teenage Cliques**! Earl Grey is crowd favorite as well as English Breakfast and perhaps an orange pekoe. There are lovely tea shops which offer fresh tea, and if you ask nicely they can provide a nice sampler pack for you. If you are a fan of history, Typhoo Tips http://wp.me/p27trL-xE was the first brand of tea offered in tea bags back in 1869, assuring customers they were buying fresh tea and not reclaimed tea.

Thank you so much!
What a delightful site; Tea and *when did garrett died*, Downton Abbey- What could be better?
Excellent information! We enjoyed the **warhol**, first episode of season three of when Downton Abby last night while partaking of High Tea!
Although I enjoy my daily cuppa (or “cuppaS”), there is nothing like a real tea ceremony to **teenage cliques**, make me feel all warm and fuzzy. I sincerely wish I had known of your website when I hosted a tea party for a few of did garrett my coworkers last year! We had cucumber sandwiches, scones with clotted cream, jam, and lemon curd, and I made “lemon drops” which are essentially slices of homemade sponge cake with lemon curd between the layers and topped with whipped cream and *vitamin c juice*, a raspberry (or a gooseberry if you’re being authentic). Such good fun!!
I am English and over did garrett morgan died the years have attended many Afternoon tea parties Including once as a girl a Royal Garden Party at Buckingham Palace , I love them. **Is A**! I am just about to host my own Downtown tea party for 60. I have found your site the most informative and well researched, it far surpasses any other research I have found.

The links to the recipes are very useful. **When Did Garrett Died**! Thank you for *Reasons and Treatments for Premature Essay* all your hard work and research. **When Did Garrett Died**! Your site is lovely.
Suzanne, Oxfordshire, Uk.
An absolutely brilliant guide. Amazing how many people confuse ‘high tea’, ‘cream tea’ and the much more substantial ‘afternoon tea’.
I’m constantly “on about” tea on my comfort food blog. Having grown up with afternoon tea, I have introduced many friends to the pleasures of afternoon tea over the years. Also love to go out for tea, and I *hate* it when servers at **stream of consciousness mrs dalloway**, even the **did garrett**, poshest places call it “high tea.” I think Americans do that because it sounds more “haute.”
Which, in *teenage cliques*, your opinion is better when it comes to a tea kettle: stainless steel or porcelain enamel?

I’m getting very tired of using a regular pot to heat water in our house but would rather invest in a quality kettle than not. Any information you could give would be appreciated. **When Did Garrett Morgan Died**! Thank you. **Stream**! #128578;
I have always used a stainless steel kettle, we currently have a smart looking brushed Cuisinart cordless version, and *did garrett died*, didn’t realize you could find porcelain, but you must live in a wonderful part of the world where they exist. **Teenage Cliques**! As for a teapot, ceramic is the way to go.
Thank you for your input!

The porcelain kettles I’ve seen are online actually. **Morgan**! Not many to choose from *mapp vs. ohio* but there are a few companies that make them. **Did Garrett Morgan Died**! Again, thank you for replying.
Well. **Andy Influenced By**! I couldn’t have found a better site to link to **did garrett**, from my article, Downton Abbey: Hats of Distinction.
As the Teapixie, I live for tea and the taking of tea. I love that special menus are created around tea and *mapp vs. ohio*, it is so fun to see how you pair recipes with Downton Abbey events. Even if the events are tragic.
Isn’t television fun? In any case, I just want to **morgan died**, let you know that I have linked to your site, along with others.

I want to invite you to come by my page so that you can see how I am profiling the fun of the **mapp vs. ohio**, Downton Abbey style-makers. I am regularly updating the article with new links and new hats.
Putting the article together is almost as fun as Tea or watching Downton Abbey, because I get to visit sites like yours. Thank you so much for *when did garrett morgan died* creating a site with true tea ambiance!
I’m terribly sorry but I really feel that I must comment. Commendable as your blog may be, I find it rather offensive when you refer to “The British may have failed miserably in other culinary areas”. **Mrs Dalloway**! I am English born and bred and I suggest you visit my fair country to actually try our food and fabulous restaurants. We have an **when**, extraordinary amount of what is a sphinx fantastic fare, amazing quality of produce, a thriving farmers market and artisan producers. **Did Garrett Morgan Died**! We have some of the best chefs in the World and thankfully independent restaurants still survive despite the ongoing march of dull franchises. As someone who was bought up in a house of food, a brother who was a pastry chef at Fortum and Mason and cooked for the royal family; I suggest you try our cuisine for yourself and on our shores before you revert to a stereotypical and spread rather outdated and uneducated view.

I think you might even enjoy the education. I wish you well.
Oh CC! My roots are in England and I do have a special place in *teenage cliques*, my heart for Jamie, Nigella, and Heston (a family favourite). **When Did Garrett Morgan Died**! Of course England is coloured by myriad gastronic experiences, just like Canada. I am on the West Coast of Canada and I rarely eat smoked salmon because it’s just too expensive! And a dish like Poutine is for those who wish to live short lives.

But we eat lots of sushi, curry, Mexican, and Italian.
It is hard to **stream mrs dalloway**, think of stereotyped British food without thinking of morgan died deep fried fish and chips, bangers and mash, and scraped toast. This is not meant as an **by**, insult to **did garrett**, the country of England, it actually gives me warm feelings about my British Grandad – even the **mapp vs. ohio**, overcooked veggies that he loved. I have eaten food on your shores, many, many times. There are fantastic restaurants and there are places that struggle to **died**, break free from the **warhol by**, historic menus.

I love both and look forward to shopping in English grocery stores, talking with restaurant and tea room prorietors, and eating a wide variety of fantastic foods including the nations number one dish, chicken tikka masala!
Please know that my personal regard for the “culinary failures” of England are associated with history – just as Afternoon tea or High Tea or Elevenses are fantastic events associated with the history. **Died**! All country cultures are weirdly stereotyped but should never be perceived as lacking in *mapp vs. ohio*, opportunities to evolve or lacking in evolution.
My connection with historic British food is **did garrett** enveloped in incredibly wonderful feelings about my own heritage. I am proud to say that I have a British culinary heritage.

So am I! I cherish all my mother passed on to me. Not just cooking but the heritage as well. **Teenage Cliques**! Rule Britiana. Joan Murphins.
I believe it is correct to say “elevenses” not “elevensies.”
in English hi tea what hot snacks we can offer.
Dear Pamela,As a Brit, it’s nice to **when morgan died**, see someone from ‘over the pond’ who’s got most of the information about Afternoon Tea correct for a change: I now live in Vinci, Italy (yes where Leonardo was born), and now offer afternoon tea to Italians in our home dining business.I would take you to task on one item in your article,(there’s always a critic!) and that is about Cream Tea in which you say: “Cream Tea — A simple tea service consisting of scones, clotted cream, marmalade or lemon curd and tea.” Cream Tea traditionally consists of Reasons for Premature Ejaculation Essay scones served with clotted cream and strawberry jam.Having said that if people prefer to have their scones (and it’s pronounced ‘skons’ as far as I’m concerned),with an alternative, I have no problem with that, it’s a free world (supposedly)!For example I sometimes fill my Victoria Sponge with lemon curd instead of the traditional raspberry jam and fresh raspberries both of when died which balance well with a nice cup of mapp vs. ohio sweet tea.Good Luck with the book!

Is Sherry served at **did garrett morgan died**, Tea?
When, before or after?
Your article is great, very helpful. **And Treatments Essay**! Thank you very much !