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  "body": "\n## Summary\n\nA cyclic algebra is a [[crossed product algebra|G21MZQCME9]] for a [[cyclic extension|VFE5B71E7Y]] of fields.\nIt is described by a generator $\\sigma$ of the Galois group and one nonzero element $a$ of the base field.\nEvery cyclic algebra of degree $2$ is a quaternion algebra, and over local and global fields every central simple algebra is cyclic.\nWhether a cyclic algebra splits is decided by the [[field norm|KRNFEZWPY0]]: it splits exactly when $a$ is a norm.\n\n## Explanations\n\n### L1 human\n\nTake a field extension $L/K$ whose symmetries form a cyclic group of order $n$ generated by $\\sigma$.\nAdd one new symbol $u$ to $L$ with two rules.\nMoving a number $x$ of $L$ past $u$ turns it into $\\sigma(x)$, and $u^n$ equals a fixed number $a$ of $K$.\nThe result is the cyclic algebra.\nFor $L = \\C$, $K = \\R$, $\\sigma$ complex conjugation and $a = -1$, this gives Hamilton's quaternions with $u = j$.\n",
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