Splitting field of a polynomial over a finite field


by resolvent1
Tags: field, finite, polynomial, splitting
resolvent1
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#1
Dec31-10, 06:41 PM
P: 24
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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Hurkyl
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#2
Dec31-10, 07:08 PM
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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
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#3
Dec31-10, 09:13 PM
P: 24
I've got it, thanks.


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