# A field of characteristic zero is perfect

How do you prove that a field F of characteristic zero is perfect, or rather that every irreducible f(x) in F[x] is separable? Thank you!

• Do you mean that every irreducible $f$ is separable? – Amitai Yuval Aug 19 '14 at 9:21
• Yes! Sorry, I shall edit the question – hiat Aug 19 '14 at 9:38

This is true because every irreducible polynomial $f(x)$ in $F[x]$ is separable (provided the characteristic of $F$ is zero, or $F^p=F$ for prime characteristic $p$). Indeed, we have $f'(x)\neq 0$ for the derivative, because $deg(f')=deg(f)-1$. Here we have used that a polynomial $f(x)$ is inseparable if and only if $f'(x)=0$.
• Why is $f(x)$ inseparable $\Leftrightarrow f'(x)=0$? Shouldn't we say that $f(x)$ is inseparable $\Leftrightarrow \gcd(f(x),f'(x))\neq 1$? – rmdmc89 May 7 at 1:00
• @rmdmc89 See this post or this post, Corollary $3.4.3$ part $(2)$. – Dietrich Burde May 7 at 8:19