Question

Hi, Let $A$ be a division algebra over a local field $F$ of dimension $n^2$ and $K$ be an extension of $F$ of degree $n$. Then if follows from COROLLARY 2 in page 225 of Weil's Basic Number Theory that $A$ is split over $K$. Now my question is

1. Does $A$ contain a subfield isomorphic to $K$ ? Why?

2. Could you descirbe the general picture about "maximal subfields in central simple algebras" over a (not necessarily local) field ? Or tell me some references.

Thank you !

Answer 1

Corollaries 3.4 and 3.7 on pages 130 and 131 of Milne's notes (jmilne.org/math/CourseNotes/CFT.pdf) will give you the answers.

Answer 2

The answer to your first question can also be found in Weil's Basic Number Theory. See Corollary 3 on page 180, and note that if two central simple algebras of the same dimension are similar then they are isomorphic.

Answer 3

To address 2.: for any central simple algebra $A$ over a field $k$, there is a well-developed theory describing the relations between finite splitting fields $l/k$ for $A$ and fields which are sub-$k$-algebras of $A$. The most important result is probably this one:

Theorem: Let $A$ be a central simple algebra over a field $k$, of dimension $n^2$. For a field extension $l/k$ of degree $n$, the following are equivalent:
(i) There is a $k$-algebra embedding $l \hookrightarrow A$.
(ii) $l$ is a splitting field for $A$.

A proof of this result can be found, for instance, in $\S 6.2$ of these notes on noncommutative algebra. Citations to more substantial treatments are given in $\S 6.10$.