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Math Help - Splitting Field

  1. #1
    ynj
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    Splitting Field

    Let Fbe a field of characteristic p. Prove that x^p-aeither is irreducible or splits in F...
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  2. #2
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    if you prove one thing you are through ...

    all roots of this polynomial are equal (in some appropriate extension)
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  3. #3
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    Quote Originally Posted by ynj View Post
    Let Fbe a field of characteristic p. Prove that x^p-a either is irreducible or splits in F...
    i'll prove something more than what the problem is asking! i'll show that either x^p -a is irreducible or x^p - a = (x + c)^p, for some c \in F:

    suppose x^p - a is not irreducible. then x^p - a = g(x)(f(x))^m, for some polynomials f(x), g(x) \in F[x], where f(x) is irreducible with 1 \leq \deg f(x) < p, \ 1 \leq m \leq p, and \gcd(f(x),g(x))=1.

    differentiating (formally) will give us: 0=px^{p-1}=g'(x)(f(x))^m + mg(x)f'(x)(f(x))^{m-1}. hence mg(x)f'(x)=-g'(x)f(x), which is impossible unless m=p, \ \deg g(x) = 0 and \deg f(x)= 1

    because \gcd(f'(x),f(x))=\gcd(f(x),g(x))=1. so suppose m=p, \ g(x) = \alpha and f(x)=\beta x + \gamma. then x^p-a = \alpha (\beta x + \gamma)^p=\alpha(\beta^p x^p + \gamma^p) and hence \alpha \beta^p = 1, \ \alpha \gamma^p = -a. letting c = \gamma \beta^{-1}

    we'll get x^p - a = (x + c)^p. \ \Box
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