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.

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