Lemma 11.5.1. Similarity.
Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over k.
Every similarity class contains a unique (up to isomorphism) finite central skew field extension of k.
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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