Splitting field of a polynomial over a finite fieldby resolvent1 Tags: field, finite, polynomial, splitting 

#1
Dec3110, 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. 



#2
Dec3110, 07:08 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

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
Dec3110, 09:13 PM

P: 24

I've got it, thanks.



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