Splitting field of a polynomial over a finite field

[resolvent1]

Homework Statement

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).

Homework Equations

Not sure.

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.

 

[Hurkyl]

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?

 

[resolvent1]

I've got it, thanks.