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
$[A : k] = [K : k]^2$,
$K$ is its own centralizer, and
$K$ 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. $\square$
Comments (0)