Lemma 11.4.2 (074A)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 11: Brauer groups
Section 11.4: Lemmas on algebras
Lemma 11.4.2 (cite)

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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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