Maximal subfield

Remarks

Every maximal subfield $L$ satisfies $[L : K] = \deg D$, the degree of $D$, and $L$ splits $D$. 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.