# Maximal subfields in a division algebra over a local field

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 !

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.

• Thanks. So any finite extension $K$ of a local field $F$ of degree $n$ could be embedded into the division algebra $A$ over $F$ of degree $n^2$. Is it right ? Commented Aug 1, 2012 at 13:35
• And in the case of global fields, when could the field $K$ be embedded into the division algebra $A$ over $F$ ? One expect $K$ and $A$ should be "compatible" locally. Commented Aug 1, 2012 at 16:27
• When $F$ is local non-Archimedean, every finite extension $K$ of $F$ of degree $n$ could be embedded into your division algebra $A$. This is contained in Remark IV.4.4(c) of Milne's notes linked to in Auguste's answer. Commented Dec 28, 2017 at 11:42

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 $$\S 6.10$$.

• Thanks very much for this nice reference for central simple algebra. I am a bit confused: don't you need to pass to an suitable $A^\prime$ similar to $A$ in (i) of the theorem ? Commented Aug 2, 2012 at 6:33
• @unknown: I don't think so (and the statement and proof of the result I quote appears in my notes). Why do you think you need to pass to $A'$? Commented Aug 2, 2012 at 7:11
• Oh,I see! I just miss the condition "of dimension $n^2$" in your statement. Sorry and thanks again. Commented Aug 2, 2012 at 7:18

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.

• Note that Weil restricts to separable-extension case for simplicity. Commented Aug 2, 2012 at 12:06