# Show that $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ is separable

how can I characterize the minimal polynomials of all elements of this extension 'to show that they have all the distinct roots?

• You can show more generally that, if $E$ is an extension field of a field $K$ of characteristic $0$, then $E$ is a separable extension. This is because $\gcd\big(p(x),p'(x)\big)=1$ for all irreducible polynomial $p(x)\in K[x]$. This is something that may fail in positive characteristics. – Batominovski Nov 12 '18 at 21:29
• This is a duplicate. – Dietrich Burde Nov 12 '18 at 21:39
• @DietrichBurde, it's not a direct duplicate of the question in the body. – lhf Nov 12 '18 at 23:11

Indeed, $$f'\ne0$$ and $$\deg f' < \deg f$$ imply $$\gcd(f,f')=1$$.
(The part that can fail in positive characteristic is $$f'\ne0$$.)