# Subfields of central division algebras over fixed global field

A generalization of this question: Let $K$ be a global field, could any finite field extension of $K$ be embedded in a finite dimensional central division algebra over $K$?

The answer is true locally. Indeed, by local class field theory, every degree $n$ central division algebra over $K$ contains every degree $n$ extension. And the answer is true for cyclic extension by previous discussion.

## 1 Answer

Yes this is true, at least if you mean $K$ is a number field. (I don't recall what is known about the function field case.) Let $L/K$ be a field extension of degree $n$, and $A/K$ be a CSA of degree $n$. The only obstruction to $L$ embedding in $K$ is that it embeds locally everywhere, i.e., one has a Hasse principle.

Pick two places $u, v$ at which $L/K$ is inert. Then, by Albert-Brauer-Hasse-Noether, there exists a (necessarily division) $A/K$ such that $A_u$ and $A_v$ have Brauer invariants $1/n$ and $-1/n$, and $A_w$ is split at all other places $w$. Then $L$ embeds in $A$ because it does locally everywhere.

This should be covered in something like Pierce's Associative Algebras or Weil's Basic Number Theory.

• I am confused. We know that the restriction map of brauer group of local fields is multiplication by extension degree, so I mean for local fields this is true. In other words, if I pick a central division algebra A over Q and consider one of its subfields L, then L_v is in A_v. – zzy Apr 30 '18 at 13:22
• @zzy Sorry I misread your question. I thought you were fixing the division algebra. I'll edit my answer. – Kimball Apr 30 '18 at 14:39