multiple roots of irreducible polynomial 2

Question

let say we have an irreducible polynomial over field $F$.

I need to prove that all roots of f have the same multiplicity.

I know that if $\text{Ch}(F)=0$ so this is easy but I don't know what to do if $Ch(F)=p>0$.

Thanks.

Answer 1

Assume $\alpha$ is a root polynomial of multplicity $\ge k$ of the polynomial $f\in F[X]$. Then $\alpha$ is also a root of the $(k-1)$th derivative $f^{(k-1)}$. Now assume $f(\beta)=0$ but $f^{(k-1)}(\beta)\ne0$. Then $f^{(k-1)}$ is not the zero polynomial and $\gcd(f,f^{(k-1)})$ is a nontrivial divisor of $f$.