# Math Help - Exercise - Irreducible Polynomial

1. ## Exercise - Irreducible Polynomial

Prove that an irreducible polynomial in $Q[x]$ has no multiples roots

Let $f(x)\in \mathbb{Q}[x]$ be irreducible. Then consider the derivative. $f'(x)$ is a polynomial of degree n-1 and since $char(\mathbb{Q})=0$, so it is not identically 0. Up to constant factors, the only factors of $f(x)$ are $f(x)$ and 1, so $f(x)$ and $f'(x)$ are relatively prime, thus $f(x)$ is separable, ie has no multiple roots.