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  "body": "\n## Summary\n\nKummer theory describes the abelian extensions of exponent dividing $n$ of a field $K$ that contains the $n$-th [[roots of unity|0YV2CKP2AV]].\nEvery such extension is obtained by adjoining $n$-th roots of elements of $K$, and these extensions correspond to subgroups of $K^\\times$ containing $K^{\\times n}$.\nThe Galois group is dual to the corresponding group of radicals through the Kummer pairing.\nKummer theory is a basic tool in [[class field theory|BEKCEYHCHX]] and in [[Iwasawa theory|Y2SPPSQ5D4]].\n\n## Explanations\n\n### L1 human\n\nLet $K$ be a field of characteristic other than $2$, so that $\\pm 1$ are two distinct square roots of unity.\nThen every extension of $K$ of degree $2$ is obtained by adjoining a square root, as with $\\Q(\\sqrt{2})$ over $\\Q$.\nKummer theory is the same idea for $n$-th roots.\nSuppose that $K$ contains $n$ distinct $n$-th roots of unity.\nThen adjoining an $n$-th root $\\sqrt[n]{a}$ gives a cyclic extension.\nAll abelian extensions whose Galois group has exponent dividing $n$ arise by adjoining several such roots.\nFor example, $\\Q(i)$ contains the fourth roots of unity, and $\\Q(i, \\sqrt[4]{2})$ is a cyclic extension of $\\Q(i)$ of degree $4$.\n",
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