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Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \hookrightarrow A$?

This is equivalent to asking whether $A \otimes_{\mathbb{Q}} K \simeq M_n(K)$. Is there a local-to-global principle that can be used?

If, for example, $A \simeq M_n(D)$ where $D$ is a quaternion algebra over $\mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?

Any reference to a similar example where this is worked out would be greatly appreciated!

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I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.

Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $\mathfrak p$ of $F$ (possibly infinite) denote by $m_\frak{p}$ the local index of $A$ at $\mathfrak{p}$ (i.e., the degree of the division algebra part of $A\otimes_F F_\mathfrak{p}$).

The following is Theorem 32.15 of Reiner.

Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $\mathfrak p$ of $F$ and prime $\mathfrak P$ of $L$ lying above $\mathfrak p$, we have: \begin{equation} m_{\mathfrak p}\mid [L_\mathfrak P:F_\mathfrak p]. \end{equation}

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  • $\begingroup$ Perhaps I am misunderstanding the definition of local index, but if I take $A/\mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_\mathfrak{p}:\mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something. $\endgroup$ – Sun Ra Feb 5 '19 at 14:52
  • $\begingroup$ @SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=\sqrt{4}=\sqrt{\dim(A\otimes_{\mathbb Q} \mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$. $\endgroup$ – user1073 Feb 5 '19 at 15:30
  • $\begingroup$ That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for. $\endgroup$ – Sun Ra Feb 5 '19 at 17:28

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