Extension fields / splitting fields proof...

Hello, put this in the homework help but think its probably best off here, Im trying to prove the following by induction....

Given a polynomial, f(X) \in F[X], of degree n, there exists

an extension field K (sub set of) F such that f(X) has n roots

in K.

This is what I did first time round...

First assume that F is irreducible and argue by induction on n.

Suppose the result holds for irreducible polynomials of degree at

most n-1. Set E= F[x]/f(x).

Now, in E, f(x) has at least one root, so we can write f(x)

= (x - \alpha_{1})(x - \alpha_{2})...(x-\alpha_{r})g(x) with

\alpha_{1}, \alpha_{2}, ..., \alpha_{r} \in E and g(x) \in

E[x] irreducible. Since deg g < n, by the inductive assumption,

the result also applies to g.

But ive been told this is wrong because there could be more than one irreducible factor of degree > 1, also I shouldnt be trying to take irreduciblity through the proof.

If someone knows how to prove this it would be much appreciated, its part of a huge project and needs to be done by tomorrow!

Cheers!

ElGamal.