(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Splitting field of a polynomial over a finite field

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