Properties of Galois Extensions

1. May 5, 2010

jk1100

1. The problem statement, all variables and given/known data
Suppose F -- K is a Galois extension and f(x) in F[x] is an irreducible
polynomial that has a root in K. Show that f(x) factors into a product
of linear factors in K[x].

2. Relevant equations
I read on Wolfram this fact is equivalent to being a Galois extension, but I am drawing a complete blank on how to show that.

3. The attempt at a solution
Idea 1:
I know that if K is an extension of F and f(x) is a polynomial with coefficients in F that any F-automorphism of K will map roots on f(x) to another roots of f(x). Now I know at least one root of f(x) is in K; so the image of that root under any F-automorphism will also be in K and will also be a root of f(x)

Idea 2:
I also know I can build a tower of fields by adjoining the element a st f(a)=0 to obtain
F -- F(a) -- K
Because the root a must be in K so that field F(a) must sit between F and K. Since F -- K is a Galois extension and F(a) is an intermediate field I know F(a) -- K must be a Galois extension as well

Idea 3:
I also know since K is an extension of F and it is Galois that then it is the splitting field for some polynomial in F[x].

I just need some way to bring these ideas together (or take one all the way). I feel like I am half way there in bunch a different ways, but I am just blanking of finishing the proof.