Lemma 11.5.1. Similarity.

1. Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over $k$.

2. Every similarity class contains a unique (up to isomorphism) finite central skew field extension of $k$.

3. If $A = \text{Mat}(n \times n, K)$ and $B = \text{Mat}(m \times m, K')$ for some finite central skew fields $K$, $K'$ over $k$ then $A$ and $B$ are similar if and only if $K \cong K'$ as $k$-algebras.

Proof. Note that by Wedderburn's theorem (Theorem 11.3.3) we can always write a finite central simple algebra as a matrix algebra over a finite central skew field. Hence it suffices to prove the third assertion. To see this it suffices to show that if $A = \text{Mat}(n \times n, K) \cong \text{Mat}(m \times m, K') = B$ then $K \cong K'$. To see this note that for a simple module $M$ of $A$ we have $\text{End}_ A(M) = K^{op}$, see Lemma 11.4.6. Hence $A \cong B$ implies $K^{op} \cong (K')^{op}$ and we win. $\square$

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