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Thread: Galois Extension

  1. #1
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    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^{*}) $.
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  2. #2
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    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.
    Last edited by ThePerfectHacker; May 29th 2008 at 04:33 PM.
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