# Galois Extension

• May 29th 2008, 04:10 PM
heathrowjohnny
Galois Extension
Let $\displaystyle L/K$ be a finite Galois extension, with Galois group $\displaystyle G$. For $\displaystyle x \in L$ set $\displaystyle \mathcal{N} x := \prod_{\sigma \in G} (\sigma x)$.

(a) Show $\displaystyle \mathcal{N} x \in K$.
(b) If $\displaystyle L$ is a finite field, show that $\displaystyle \mathcal{N}$ is the map $\displaystyle x \mapsto x^i$, where $\displaystyle i = (L^{*} : K^{*})$.
• May 29th 2008, 04:21 PM
ThePerfectHacker
For (a) show that $\displaystyle N(x)$ is fixed for all $\displaystyle \sigma \in G$* which means $\displaystyle N(x) \in L^G$ but $\displaystyle L^G = F$ because it is Galois.

For (b) since $\displaystyle |L| < \infty$ it means $\displaystyle \text{Gal}(L/F) = \left< \theta \right>$ where $\displaystyle \theta$ is the Frobenius automorphism, this is enough to prove this result.

*)This is because if $\displaystyle G$ is a group with elements $\displaystyle \{ a_1,...,a_n\}$ then $\displaystyle \{ \sigma a_1,...,\sigma a_n\}$ is a permutation.