1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Splitting field of a polynomial over a finite field

  1. Dec 31, 2010 #1
    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. jcsd
  3. Dec 31, 2010 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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?
     
  4. Dec 31, 2010 #3
    I've got it, thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Splitting field of a polynomial over a finite field
Loading...