Lemma 11.4.2. Let $A$ be a finite simple $k$-algebra. Then the center $k'$ of $A$ is a finite field extension of $k$.

Proof. Write $A = \text{Mat}(n \times n, K)$ for some skew field $K$ finite over $k$, see Theorem 11.3.3. By Lemma 11.4.1 the center of $A$ is $k \otimes _ k k'$ where $k' \subset K$ is the center of $K$. Since the center of a skew field is a field, we win. $\square$

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