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Math Help - Frobenius map

  1. #1
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    Frobenius map

    I need help on this problem





    Thanks
    Last edited by capitalX; June 24th 2008 at 10:17 PM.
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  2. #2
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    Quote Originally Posted by capitalX View Post
    I need help on this problem

    Let R be a commutative ring of prime characteristic p. Show that the Frobenius map x->x^p is a ring homomorphism from R to R.

    Thanks
    i expect someone who's studying algebra knows this much from elementary number theory that for any prime

    number p and integer 0 < k < p, we have: p \ | \binom{p}{k}. so: \binom{p}{k} \alpha = 0, \ \forall \alpha \in R. hence, since R is commutative,

    for any a,b \in R: \ (a+b)^p=a^p+b^p + \sum_{k=1}^{p-1} \binom{p}{k}a^kb^{p-k}=a^p+b^p. besides it's obvious that (ab)^p=a^pb^p.

    therefore the map x \rightarrow x^p is a ring homomorphism. Q.E.D.
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