Theorem 59.61.1. Let $K$ be a field. For a unital, associative (not necessarily commutative) $K$-algebra $A$ the following are equivalent

1. $A$ is finite central simple $K$-algebra,

2. $A$ is a finite dimensional $K$-vector space, $K$ is the center of $A$, and $A$ has no nontrivial two-sided ideal,

3. there exists $d \geq 1$ such that $A \otimes _ K \bar K \cong \text{Mat}(d \times d, \bar K)$,

4. there exists $d \geq 1$ such that $A \otimes _ K K^{sep} \cong \text{Mat}(d \times d, K^{sep})$,

5. there exist $d \geq 1$ and a finite Galois extension $K'/K$ such that $A \otimes _ K K' \cong \text{Mat}(d \times d, K')$,

6. there exist $n \geq 1$ and a finite central skew field $D$ over $K$ such that $A \cong \text{Mat}(n \times n, D)$.

The integer $d$ is called the degree of $A$.

Proof. This is a copy of Brauer Groups, Lemma 11.8.6. $\square$

Comment #7272 by Yijin Wang on

Typo in theorem 59.61.1(5): A otimes_｛K’｝K’ =Mat(d\times d，K’) should be A otimes_｛K｝K’ =Mat(d\times d，K’)

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