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  "body": "\n## Definition\n\nLet $L/K$ be a [[cyclic extension|VFE5B71E7Y]] of degree $n$, and let $\\sigma$ be a generator of its Galois group.\nLet $a \\in K$ be nonzero.\nThe *cyclic algebra* $(L/K, \\sigma, a)$ is the $L$-vector space with basis $1, u, u^2, \\dots, u^{n-1}$, with the multiplication determined by\n$$u \\, x = \\sigma(x) \\, u \\quad (x \\in L), \\qquad u^n = a.$$\n\n## Remarks\n\nIt is the [[crossed product algebra|G21MZQCME9]] for the factor set $f(\\sigma^i, \\sigma^j) = 1$ if $i + j \u003c n$ and $f(\\sigma^i, \\sigma^j) = a$ if $i + j \\ge n$, for $0 \\le i, j \u003c n$.\nSo it is a [[central simple algebra|5HRKJBAY2V]] over $K$ of degree $n$, split by $L$.\nIt is isomorphic to $M_n(K)$ if and only if $a$ is a [[norm|KRNFEZWPY0]] from $L$ to $K$, that is, $a = N_{L/K}(b)$ for some $b \\in L$.\n\nExample: $L = \\C$, $K = \\R$, $\\sigma$ complex conjugation and $a = -1$ give $\\mathbb{H}$.\nSince $N_{\\C/\\R}(z) = \\abs{z}^2 \\ge 0$, the number $-1$ is not a norm, and $\\mathbb{H}$ is not split, as expected.\nWith $a = 1 = N_{\\C/\\R}(1)$ one gets $M_2(\\R)$.\n\nNon-example: if $L/K$ is Galois but not cyclic, such as $\\Q(\\sqrt{2}, \\sqrt{3})/\\Q$, there is no single generator $\\sigma$, and the construction does not apply.\n\nIf the characteristic of $K$ is not $2$, the quaternion algebra $\\left(\\frac{a, b}{K}\\right)$ is the cyclic algebra $(K(\\sqrt{a})/K, \\sigma, b)$ when $a$ is not a square in $K$.\n",
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