# Splitting field of a polynomial over a finite field

1. Dec 31, 2010

### resolvent1

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

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

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 $$F[x]/(f) \cong F(\alpha)$$ for some $$\alpha$$. Then the substitution homomorphism ($$g \rightarrow g(\alpha)$$) might help, if I knew that $$\alpha$$ is a root of f.

Thanks in advance.

2. Dec 31, 2010

### Hurkyl

Staff Emeritus
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?

3. Dec 31, 2010

### resolvent1

I've got it, thanks.

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