Abstract Algebra: x^p-a irreducible using automorphisms

[qftqed]

Homework Statement

Let F be a field with p$\in$N, a prime natural number. Show that either X$^{p}$-$\alpha$ is irreducible in F[X] or $\alpha$ has a pth root in F

Homework Equations

The Attempt at a Solution

I'm trying to do this without making reference to the field norm, so far I've come up with an incomplete proof that is missing some key components. First, if $\alpha$ has a pth root in F then the polynomial (call it f) it is trivially reducible in F[X]. Assume then that there is no such thing in F. Assume also for a contradiction that f=gh, where g and h are not linear. There exists an extension field E in which f is a product of linear polynomials. So f=gh=(X-a$_{1}$)(X-a$_{2}$)...(X-a$_{p}$). Clearly, some of these linear factors will be factors of g and the rest factors of h. Suppose (X-a$_{i}$) is a factor of g and (X-a$_{j}$) not a factor of g. I want to say that there exists an F-automorphism (i.e. one that fixes F pointwise) that will take a$_{i}$ to a$_{j}$. The coefficients in g will be unchanged by this automorphism, while the factors of g will not, suggesting that the field F is not a Unique Factorization Domain. This would be a contradiction and so f is irreducible. My issue is that I'm not sure whether such an automorphism necessarily exists and, if it does, what any of this has to do with p being prime, which I'm sure should be a key point in this proof. Any help would be much appreciated!

 

[micromass]

Let $K$ be a splitting field of the polynomial $f$.

Can you first show that every root of $f$ is distinct?

Second, can you show that $K$ contains all $p$th root of unity?

Can you show then that if $\zeta$ is a $p$th root of unity (which is necessarily in $K$) and if $\alpha$ is a root of $f$, then all roots of $f$ are given by

$$\{\alpha\zeta^n~\vert~0\leq n< p \}$$

Then if we can write $f(X) = g(X)h(X)$ with $g$ and $h$ polynomials with coefficients in $F$, show that we can write

$$g(X) = \prod_{n\in S} (X- \alpha \zeta^n)~\text{and}~h(X) = \prod_{n\in S^c} (X-\alpha\zeta^n)$$

The constant term of $g$ is in $F$, what is this constant term? Try to deduce that a root of $f$ lies in $F$.