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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.\nThe dimension $\\dim_K A$ is a perfect square $n^2$ with $n \\ge 1$.\nThe *degree* of $A$ is the integer $n = \\sqrt{\\dim_K A}$, written $\\deg A$.\n\n## Remarks\n\nLet $\\bar{K}$ be an [[algebraic closure|0X6AB5P3G1]] of $K$.\nThe [[tensor product|1735VGRT2P]] $A \\otimes_K \\bar{K}$ is central simple over $\\bar{K}$, hence isomorphic to $M_n(\\bar{K})$ for some $n$ by Wedderburn's theorem.\nSo $\\dim_K A = \\dim_{\\bar{K}} M_n(\\bar{K}) = n^2$ is a square.\n\nExample: $\\deg M_n(K) = n$, and the quaternion algebra $\\mathbb{H}$ over $\\R$ has degree $2$, since $\\dim_\\R \\mathbb{H} = 4$.\n\nNon-example: $\\C$ has dimension $2$ over $\\R$, which is not a square, so it has no degree as an algebra over $\\R$; it is not central over $\\R$.\n\nIf $A \\cong M_m(D)$ with $D$ a division algebra, then $\\deg A = m \\deg D$.\nDegree is multiplicative: $\\deg(A \\otimes_K B) = \\deg A \\cdot \\deg B$.\n",
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