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Central simple algebras are classified by Galois cohomology of PGL.

Lemma 59.61.5. Let $K$ be a field and let $K^{sep}$ be a separable algebraic closure. Then the set of isomorphism classes of central simple algebras of degree $d$ over $K$ is in bijection with the non-abelian cohomology $H^1(\text{Gal}(K^{sep}/K), \text{PGL}_ d(K^{sep}))$.

Sketch of proof.. The Skolem-Noether theorem (see Brauer Groups, Theorem 11.6.1) implies that for any field $L$ the group $\text{Aut}_{L\text{-Algebras}}(\text{Mat}_ d(L))$ equals $\text{PGL}_ d(L)$. By Theorem 59.61.1, we see that central simple algebras of degree $d$ correspond to forms of the $K$-algebra $\text{Mat}_ d(K)$. Combined we see that isomorphism classes of degree $d$ central simple algebras correspond to elements of $H^1(\text{Gal}(K^{sep}/K), \text{PGL}_ d(K^{sep}))$. For more details on twisting, see for example [SilvermanEllipticCurves]. $\square$

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Comment #3247 by Giulio on

Suggested slogan: Central simple algebras are classified by Galois cohomology of PGL.

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