Every irreducible polynomial in $\mathbb{F}_p[x]$ is separable?

Question

How can I show this?

I tried proving the contrapositive statement but didn't get anywhere.

I think I may have to do something involving automorphisms of the splitting field of such a polynomial, and showing that the existence of repeated roots gives a contradiction. Is this along the right track?

Answer 1

If $f$ is inseparable, $f \in F_p[X^p]$

Suppose $f=a_0+a_1X^p+...+a_nX^{np}$. In finite field $F_p$, $a_i^p=a_i$.

Then $f=a_0^p+a_1^pX^p+...+a_n^pX^{np}=(a_0+a_1X+...+a_nX^n)^p$.

So $f$ is not irreducible. Contradiction.