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Math Help - surjective function

  1. #1
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    surjective function

    if N is a norm function from Fq^n to Fq . how can I prove that for alpha belongs to Fq^n star , N(alpha) =1 iff there exists an element Beta belongs to Fq^n such that alpha is equal to (Beta)^1-q .
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  2. #2
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    Quote Originally Posted by Mike12 View Post
    if N is a norm function from Fq^n to Fq . how can I prove that for alpha belongs to Fq^n star , N(alpha) =1 iff there exists an element Beta belongs to Fq^n such that alpha is equal to (Beta)^1-q .

    It's hard to understand what you mean (learn to use LaTeX! In the forum' section

    "Help with LaTeX"), but I think that what you want is Hilbert's Theorem 90, since \mathbb{F}_{q^n}/\mathbb{F}_q is cyclic

    of degree n and its Galois group is generated by the Frobenius map \phi(x):=x^q .

    Under these conditions Hibert's Theorem 90 says that \alpha\in\mathbb{F}_{q^n} has norm 1 iff

    \exists 0\neq \beta\in\mathbb{F}_{q^n}\,\,s.t.\,\,\alpha=\beta\p  hi(\beta)^{-1} .

    Now, I'm not sure but it seems this is what you want. You can either google the result or look for it

    in any decente Galois Theory book.

    Tonio
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