[Math] Number of distinct roots of an irreducible polynomial divides the degree of the polynomial

Question

Let $f(X)$ be an irreducible polynomial in the polynomial ring $k[X]$ over the field $k$. Prove that the number of distinct roots of $f(X)$ divides the degree of $f(X)$.

Answer 1

Irreducible polynomials over fields of characteristic zero have distinct roots (in the splitting field), so it suffices to consider the case where $k$ has positive characteristic. Fix some $f$ over such a $k$. We may assume $f$ has repeated roots, or the result is trivial.

The multiple roots are roots of $\gcd(f,f')$. Since $f$ is irreducible, it must divide $f'$, and this is only possible if $f'=0$, so $f(X)=g(X^p)$, where $p$ is the characteristic of $k$. We may continue this process to write $f(X)=h(X^{{p^e}})$, where $e\ge1$ and $h$ is irreducible and separable. Hence every root of $f$ has the same multiplicity $p^e$, and the result follows.