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  "body": "\n## Summary\n\nA cyclic extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is [[cyclic|0B85Q41RP6]].\nCyclic extensions are the building blocks of radical extensions.\nIf a field contains $n$ distinct $n$-th roots of unity, adjoining an $n$-th root of one of its elements gives a cyclic extension.\n[[Kummer theory|XZQENYV15Y]] and [[Artin–Schreier theory|7YNMNRGH3T]] describe cyclic extensions of a given degree.\n\n## Explanations\n\n### L1 human\n\nA cyclic extension has one basic symmetry, and every other symmetry is a repeat of it.\nEvery finite extension of finite fields is cyclic, generated by the map $x \\mapsto x^q$, where $q$ is the size of the smaller field.\nFor example, $\\Q(\\sqrt{5})$ over $\\Q$ is cyclic of degree $2$, generated by $\\sqrt{5} \\mapsto -\\sqrt{5}$.\nEvery cyclic extension is abelian, but $\\Q(\\sqrt{2}, \\sqrt{3})$ over $\\Q$ is abelian and not cyclic.\n",
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