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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nA central simple algebra over $K$ is a finite-dimensional associative $K$-algebra $A \\ne 0$ whose center is $K$ and whose only two-sided ideals are $0$ and $A$.\nFor a $K$-algebra $A$, let $M_m(A)$ be the [[matrix ring|0FHVPRAE60]] of $m \\times m$ matrices over $A$.\nTwo central simple algebras $A$ and $B$ over $K$ are *Brauer equivalent* if $M_m(A) \\cong M_n(B)$ as $K$-algebras for some integers $m, n \\ge 1$.\nThe *Brauer group* $\\mathrm{Br}(K)$ is the set of Brauer equivalence classes $[A]$, with the operation\n$$[A] \\cdot [B] = [A \\otimes_K B],$$\nthe [[tensor product|F0HQXQ0HCD]] over $K$.\n\n## Remarks\n\nThe operation is well defined, and $\\mathrm{Br}(K)$ is an [[abelian group|071DDFVG4J]] with identity $[K]$ and inverse $[A]^{-1} = [A^{\\mathrm{op}}]$, the opposite algebra; this rests on $A \\otimes_K A^{\\mathrm{op}} \\cong M_d(K)$ with $d = \\dim_K A$.\nBy Wedderburn's theorem each central simple algebra is $M_n(D)$ for a central division algebra $D$ unique up to isomorphism, so each class contains exactly one such $D$.\nExample: $\\mathrm{Br}(\\R)$ has two elements, $[\\R]$ and $[\\mathbb{H}]$, by the [[Frobenius theorem|QDJ7PQE9SW]].\nExample: $\\mathrm{Br}(K) = 0$ for $K$ algebraically closed and for $K$ finite.\nNon-example: the $K$-algebra $K \\times K$ has no class in $\\mathrm{Br}(K)$, since its center is $K \\times K$, not $K$, and $K \\times \\set{0}$ is a two-sided ideal.\nThere is an isomorphism $\\mathrm{Br}(K) \\cong H^2(K, (K^s)^\\times)$ with [[Galois cohomology|9MPCZ3EAJM]], where $K^s$ is a separable closure of $K$.\n",
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