Lemma 11.8.6. Let $k$ be a field. For a $k$-algebra $A$ the following are equivalent

$A$ is finite central simple $k$-algebra,

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

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

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

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

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

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

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