# Cyclotomic fields and splitting of central simple algebras

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!

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: $$$$m_{\mathfrak p}\mid [L_\mathfrak P:F_\mathfrak p].$$$$

• 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. Feb 5, 2019 at 14:52
• @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 $M_n(D)$.
– user1073
Feb 5, 2019 at 15:30
• That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for. Feb 5, 2019 at 17:28