Splitting field of a polynomial over a finite field

  1. 1. The problem statement, all variables and given/known data
    Assume F is a field of size p^r, with p prime, and assume [tex]f \in F[x][/tex] is an irreducible polynomial with degree n (with both r and n positive).

    Show that a splitting field for f over F is [tex]F[x]/(f)[/tex].

    2. Relevant equations
    Not sure.

    3. The attempt at a solution
    I know from Kronecker's theorem that f has a root in some extension field of F, but I don't know that this root is necessarily in F[x]/(f). If I could obtain this, I could use the fact that finite extensions of finite fields are Galois, therefore normal (and separable), so f splits in F[x]/(f).
    I also know that finite extensions of finite fields are simple, so [tex]F[x]/(f) \cong F(\alpha)[/tex] for some [tex]\alpha[/tex]. Then the substitution homomorphism ([tex]g \rightarrow g(\alpha)[/tex]) might help, if I knew that [tex]\alpha[/tex] is a root of f.

    Thanks in advance.
  2. jcsd
  3. Hurkyl

    Hurkyl 15,987
    Staff Emeritus
    Science Advisor
    Gold Member

    I think you're thinking too hard. You want to find some polynomial expression in x that you can plug into f to get something that is divisible by f(x), right?
  4. I've got it, thanks.
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