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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]].\nA *central simple algebra* over $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ over $K$ with these three properties:\n\n1. $A$ is finite-dimensional as a vector space over $K$.\n2. $A$ is a [[simple ring|62M0NC3E2T]]: $A \\ne 0$ and its only two-sided ideals are $0$ and $A$.\n3. The [[center|MVT64B3B6H]] of $A$ is $K \\cdot 1$.\n\n## Remarks\n\nExample: $M_n(K)$ is a central simple algebra over $K$ for $n \\ge 1$.\nIt has dimension $n^2$, its only two-sided ideals are $0$ and itself, and a matrix that commutes with all matrix units $E_{ij}$ is a scalar matrix.\n\nNon-example: $\\C$ as an algebra over $\\R$ is finite-dimensional and simple, but its center is $\\C \\ne \\R$, so clause 3 fails.\nNon-example: $K \\times K$ is commutative of dimension $2$ and has the ideal $K \\times 0$, so clause 2 fails.\n\nHamilton's quaternions $\\mathbb{H}$ are a central simple algebra over $\\R$ of dimension $4$.\nThe dimension of a central simple algebra is always a square $n^2$, and $n$ is its degree.\nIf $A$ and $B$ are central simple over $K$, so is $A \\otimes_K B$; and $A \\otimes_K A^{\\mathrm{op}} \\cong M_{n^2}(K)$ when $\\dim_K A = n^2$.\n",
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