---
tag: ZSVF54TCYP
kind: concept
name: Maximal subfield
msc: [16K20, "12E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
---

## Summary

Let $D$ be a [[division ring|0EA5BTA0CX]] that is finite-dimensional over its center, a field $K$.
A maximal subfield of $D$ is a subfield that contains $K$ and is not contained in a larger subfield.
Its degree over $K$ always equals the [[degree|NNXSP5KKAC]] of $D$, and it [[splits|NVD7R0T835]] $D$.
So the study of a division algebra reduces in part to the study of the fields it contains.

## Explanations

### L1 human

A division algebra contains many commutative pieces, and some of them are fields.
A maximal subfield is one of these fields that cannot be made larger inside the algebra.
In Hamilton's quaternions over $\R$, the elements $x + yi$ form a copy of $\C$, which is a maximal subfield.
There are infinitely many such copies, one for each line through $0$ in the space spanned by $i, j, k$.