= Descent classification of central simple algebras
{title2=$\operatorname{CSA}_n(L/K)\cong H^1(G,\operatorname{PGL}_n(L))$}
A splitting isomorphism of a <central simple algebra> with $M_n(L)$ transports its Galois action to $T_\sigma=c_\sigma\sigma_0$, giving a <nonabelian first cohomology> class. Conversely a cocycle gives a semilinear algebra action whose invariants descend to a <central simple algebra>. Equivalent cocycles give isomorphic fixed algebras. This gives a pointed-set bijection $\operatorname{CSA}_n(L/K)\cong H^1(\operatorname{Gal}(L/K),\operatorname{PGL}_n(L))$.
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