How to Calculate Norms of Field Extensions in Galois Theory

[Ypsilon IV]

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!

 

[adriank]

Well, if E/F is a Galois extension, then the norm satisfies
$$N_{E/F}(u) = \prod_{\varphi \in \operatorname{Gal}(E/F)} \varphi(u) \quad(u \in E).$$

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