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Abstract Algebra: x^p-a irreducible using automorphisms

  1. May 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Let F be a field with p[itex]\in[/itex]N, a prime natural number. Show that either X[itex]^{p}[/itex]-[itex]\alpha[/itex] is irreducible in F[X] or [itex]\alpha[/itex] has a pth root in F


    2. Relevant equations



    3. The attempt at a solution
    I'm trying to do this without making reference to the field norm, so far I've come up with an incomplete proof that is missing some key components. First, if [itex]\alpha[/itex] has a pth root in F then the polynomial (call it f) it is trivially reducible in F[X]. Assume then that there is no such thing in F. Assume also for a contradiction that f=gh, where g and h are not linear. There exists an extension field E in which f is a product of linear polynomials. So f=gh=(X-a[itex]_{1}[/itex])(X-a[itex]_{2}[/itex])...(X-a[itex]_{p}[/itex]). Clearly, some of these linear factors will be factors of g and the rest factors of h. Suppose (X-a[itex]_{i}[/itex]) is a factor of g and (X-a[itex]_{j}[/itex]) not a factor of g. I want to say that there exists an F-automorphism (i.e. one that fixes F pointwise) that will take a[itex]_{i}[/itex] to a[itex]_{j}[/itex]. The coefficients in g will be unchanged by this automorphism, while the factors of g will not, suggesting that the field F is not a Unique Factorization Domain. This would be a contradiction and so f is irreducible. My issue is that I'm not sure whether such an automorphism necessarily exists and, if it does, what any of this has to do with p being prime, which I'm sure should be a key point in this proof. Any help would be much appreciated!
     
  2. jcsd
  3. May 19, 2014 #2

    micromass

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    Let ##K## be a splitting field of the polynomial ##f##.

    Can you first show that every root of ##f## is distinct?

    Second, can you show that ##K## contains all ##p##th root of unity?

    Can you show then that if ##\zeta## is a ##p##th root of unity (which is necessarily in ##K##) and if ##\alpha## is a root of ##f##, then all roots of ##f## are given by

    [tex]\{\alpha\zeta^n~\vert~0\leq n< p \}[/tex]

    Then if we can write ##f(X) = g(X)h(X)## with ##g## and ##h## polynomials with coefficients in ##F##, show that we can write

    [tex]g(X) = \prod_{n\in S} (X- \alpha \zeta^n)~\text{and}~h(X) = \prod_{n\in S^c} (X-\alpha\zeta^n)[/tex]

    The constant term of ##g## is in ##F##, what is this constant term? Try to deduce that a root of ##f## lies in ##F##.
     
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