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I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's concentrate on these:

  1. Which number fields $K$ occur as subfields of a finite-dimensional division algebra over $\mathbb{Q}$ with center $\mathbb{Q}$?

  2. Which pairs of number fields $K, L$ occur as subfields of the same finite-dimensional division algebra over $\mathbb{Q}$ with center $\mathbb{Q}$?

There are some easy examples involving quaternions but I am curious how completely these kinds of questions are understood. Some preliminary Googling on my part was not successful.

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I think you want $\mathbb{Q}$ in the title (instead of Q) –  Belgi Aug 5 '12 at 4:58
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@Belgi: I don't know. $\LaTeX$ in titles takes time to display and I'd prefer not to slow down the main page. I am also not sure what using the $\LaTeX$ would do to the searchability of the title. –  Qiaochu Yuan Aug 5 '12 at 4:59
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+1 A very good question. May be class field theory offers an answer? By the construction here all the cyclic extensions occur, but that is nowhere near a complete answer to your 1st question. Undoubtedly you found that quaternions contain all the fields $\mathbb{Q}(\sqrt{-d})$, where $d$ is a some of three squares. –  Jyrki Lahtonen Aug 5 '12 at 5:23
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@Andrea: That last sentence was a comment to question 2. Hamilton's quaternion $ai+bj+ck$ is a square root of $-a^2-b^2-c^2$, so $\mathbf{H}$ contains all those fields. The fact that all quadratic extensions occur in some division algebra was already covered by my first comment (quadratic extensions being cyclic). –  Jyrki Lahtonen Aug 5 '12 at 9:48
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I asked a colleague. A degree $n$ extension field splits the the division algebra $D$, iff it is isomorphic to a maximal subfield of $D$. Also the question, whether a field splits $D$, can be decided locally by studying the behavior of Hasse invariants under extension of scalars. I am still working to understand the description of that in a way that I could communicate. Hopefully one of the resident class field theorists shows up soon. –  Jyrki Lahtonen Aug 6 '12 at 11:31

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