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if E is a splitting field for a division algebra D, is it always true that E can be embedded in D? Jacobson (BA II, Th. 4.8) only states that E can be embedded into Mat(r,D) for some r.

flag	asked Oct 5 at 1:22 George 53●2

BR · Accepted Answer · 2012-10-05 01:37:41Z UTC

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It depends on what you mean by "splitting field". What is true is that for a division algebra $D$ (over $F$) of dimension $n^2$, if $E$ is a field extension of degree $n$ over $F$ that splits $D$, then $E$ is isomorphic to a subfield of $D$. See the Theorem on page 16 of Paul Garrett's notes.

Obviously, an algebraic closure of $F$ will split $D$, but won't be isomorphic to a subfield for dimension-counting reasons.

answered Oct 5 at 1:37

BR
2,317●1●6●14

Thanks for the reference. – George Oct 5 at 1:57

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Or flip ahead several pages to read Theorem 4.12 in BA II... – grp Oct 5 at 3:41

Are all such fields equivalent? All I can tell from the references, they have the same [E:F] = deg(D). – George Oct 5 at 10:55

I asked this as a separate question: mathoverflow.net/questions/108907/… – George Oct 5 at 12:12

George, just so you know, it is ok to ask for clarification to an answer in the comments; you don't necessarily need to ask it as a separate question. A quaternion algebra over $\mathbb Q$ is split by any quadratic extension of $\mathbb Q$ that does not split at the places the quaternion algebra is ramified. E.g., if a quaternion algebra is ramified at $\infty$ and $2$, then any imaginary quadratic extension of $\mathbb Q$ such that $2$ does not split in the extension will split the quaternion algebra. – BR Oct 5 at 12:37

splitting field for a division algebra

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