How to Calculate Norms of Field Extensions in Galois Theory

In summary, the conversation discusses the topic of finding norms in field extensions. The participant mentions feeling comfortable representing norms as determinants of linear operators but needing help with representing norms as products of isomorphisms. They have read a textbook but would like to see an example. They request guidance on finding the norm for a specific field extension. The response mentions that if the extension is Galois, the norm can be found by taking the product of the images of an element under all automorphisms in the Galois group. The participant is asked to write out the elements of the Galois group for a specific example.
  • #1
Ypsilon IV
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Hello everyone, I need some help with finding norms of the field extension.

I feel pretty comfortable when representing norms as determinants of linear operators but I seem to be stuck with representing norms as product of isomorphims.

I have read Lang's GTM Algebra, but I really would like to see an example, how it works.

I would really appreciate if someone could help me with the guidelines how to find the norm for say Q(sqrt(2), sqrt(3)). Thanks in advance!
 
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  • #2
Well, if E/F is a Galois extension, then the norm satisfies
[tex]N_{E/F}(u) = \prod_{\varphi \in \operatorname{Gal}(E/F)} \varphi(u) \quad(u \in E).[/tex]

So write out the elements of the Galois group Gal(Q(√2, √3)/Q). (There are four of them.)
 

What is a norm of a field extension?

A norm of a field extension is a function that maps elements from a larger field extension to a smaller base field. It is used to measure the size or magnitude of elements in the larger field extension.

What is the purpose of a norm in a field extension?

The purpose of a norm in a field extension is to provide a way to compare and measure elements in a larger field extension. It is also used in various algebraic and number theory applications.

How is a norm calculated in a field extension?

The specific method for calculating a norm in a field extension depends on the specific field extension and the chosen norm function. In general, a norm can be calculated by multiplying the element in the larger field extension by its conjugates (elements with the same norm) and taking the product of all these values.

What is the relationship between a norm and a field extension degree?

The norm of an element in a field extension is related to the degree of the field extension. Specifically, the norm of an element raised to the power of the degree of the field extension is equal to the product of all the conjugates of that element.

What are some applications of norms in field extensions?

Norms in field extensions have various applications in algebraic and number theory. They are used in Galois theory, quadratic forms, and cyclotomic fields, among others. They also have applications in coding theory and cryptography.

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