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Math Help - Galois Extension

  1. #1
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    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^{*}) .
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  2. #2
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    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.
    Last edited by ThePerfectHacker; May 29th 2008 at 05:33 PM.
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