I know that irreducible polynomials over fields of zero characteristic have distinct roots in its splitting field.
Theorem 7.3 page 27 seems to show that irreducible polynomials over $\Bbb F_p$ have distinct roots in its splitting field (and all the roots are powers of one root). Is the proof correct? I have never seen this result anywhere else. The proof is very convincing to me.
Does the result hold for $\Bbb F_q$ where q is a power of prime? I don't think it holds because I've heard there are irreducible polynomials with repeated roots?