Show that a polynomial is irreducible but not separable over a particular field

Hi, this was a problem on an exam, and I wasn't sure how to do it.

The problem is as follows:

Let , where is prime. Show that is irreducible in .

I'm lost on this one. I feel like I should assume that it's reducible, which implies that can be factored into irreducibles of lesser degree, but somehow there aren't any products of irreducibles that give us . Unfortunately, I don't know how to show the "somehow" part :)

Hope someone can help! Thanks!

Re: Show that a polynomial is irreducible but not separable over a particular field

you do know that in a field F with char(F) = p, that:

, right?

in particular, is the ONLY root of , in any extension of .

so it suffices to show .

for suppose it were. then we would have:

for some polynomials .

suppose deg(p) = k, and deg(q) = m. then deg(p(x)^{p}) = kp, and deg(q(x)^{p}) = mp.

thus we have:

1 + mp = kp

but p divides kp, and cannot divide 1 + mp. so we have our contradiction.

Re: Show that a polynomial is irreducible but not separable over a particular field

Ah, yes! I do know that, I just didn't make the connection that the pth root will be the only root. Thank you!!