---
tag: 7Z0XM9N43F
kind: definition
name: Maximal subfield
msc: [16K20, "12E15"]
defines: ZSVF54TCYP
uses: [5HRKJBAY2V, 0EA5BTA0CX, 0GFGHRT2FJ, NNXSP5KKAC, NVD7R0T835]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
---

## Definition

Let $K$ be a field and let $D$ be a [[division ring|0EA5BTA0CX]] that is a [[central simple algebra|5HRKJBAY2V]] over $K$.
A [[subfield|0GFGHRT2FJ]] of $D$ is a subring $L$ of $D$ that is a field.
A *maximal subfield* of $D$ is a subfield $L$ with $K \subseteq L$ that is not properly contained in any other subfield of $D$.

## Remarks

Every maximal subfield $L$ satisfies $[L : K] = \deg D$, the [[degree|NNXSP5KKAC]] of $D$, and $L$ [[splits|NVD7R0T835]] $D$.
Indeed, maximality gives $C_D(L) = L$, and the double centralizer theorem gives $[L : K]^2 = \dim_K D$.
Conversely, a subfield $L \supseteq K$ with $[L : K] = \deg D$ is maximal.
For a central simple algebra $A$ that is not a division algebra, a subfield $L \supseteq K$ of $A$ with $[L : K] = \deg A$ is also called a maximal subfield.

Example: in $\mathbb{H}$ over $\R$, the subalgebra $L = \R + \R i$ is a field isomorphic to $\C$, and $[L : \R] = 2 = \deg \mathbb{H}$, so $L$ is a maximal subfield.

Non-example: the subfield $\R$ of $\mathbb{H}$ contains $\R$ but is properly contained in $\R + \R i$, so it is not maximal.