# The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ over $\mathbb{Q}$? Two necessary conditions are $\mathrm{lcm}(n,m) \mid d$ and $d \leq nm$. (In particular, if $n,m$ are coprime, only $d=nm$ is possible.) Are they sufficient? Or do we actually have $d \mid nm$? I have chosen $\mathbb{Q}$ just to fix ideas, maybe the same analysis works for any field (of characteristic zero). So an answer treating this more general case is appreciated as well.

• $\newcommand{\Q}{\mathbb{Q}}$ A quick observation: If either $\Q(a)$ or $\Q(b)$ is Galois over $\Q$ then we do have $d \vert nm$, but that is not true in general: the degree of $\Q(\sqrt{2}, \zeta_3\sqrt{2})$ over $\Q$ is $6$ even though those two algebraic integers each have degree $3$ over $\Q$. Jul 15, 2017 at 14:47
• Here is the mathoverflow version. Oct 31, 2017 at 17:44