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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.

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    $\begingroup$ $\newcommand{\Q}{\mathbb{Q}}$ 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 $\Q(\sqrt[3]{2}, \zeta_3\sqrt[3]{2})$ over $\Q$ is $6$ even though those two algebraic integers each have degree $3$ over $\Q$. $\endgroup$
    – user263190
    Jul 15, 2017 at 14:47
  • $\begingroup$ Here is the mathoverflow version. $\endgroup$
    – HeinrichD
    Oct 31, 2017 at 17:44

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