Maximal subfield

Summary

Let $D$ be a division ring 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 of $D$, and it splits $D$. So the study of a division algebra reduces in part to the study of the fields it contains.