Lemma 11.7.3. Let $A$ be a finite central simple algebra over $k$. If $K \subset A$ is a subfield, then the following are equivalent

1. $[A : k] = [K : k]^2$,

2. $K$ is its own centralizer, and

3. $K$ is a maximal commutative subring.

Proof. Theorem 11.7.1 shows that (1) and (2) are equivalent. It is clear that (3) and (2) are equivalent. $\square$

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