# Galois Extension

• May 29th 2008, 05:10 PM
heathrowjohnny
Galois Extension
Let $L/K$ be a finite Galois extension, with Galois group $G$. For $x \in L$ set $\mathcal{N} x := \prod_{\sigma \in G} (\sigma x)$.

(a) Show $\mathcal{N} x \in K$.
(b) If $L$ is a finite field, show that $\mathcal{N}$ is the map $x \mapsto x^i$, where $i = (L^{*} : K^{*})$.
• May 29th 2008, 05:21 PM
ThePerfectHacker
For (a) show that $N(x)$ is fixed for all $\sigma \in G$* which means $N(x) \in L^G$ but $L^G = F$ because it is Galois.

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

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