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  "body": "\n## Summary\n\nAn abelian extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is abelian.\nBy the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], every intermediate field of a finite abelian extension is again Galois over the base.\n[[Cyclic extensions|VFE5B71E7Y]] and [[cyclotomic extensions|VD6VC3ZMMY]] of $\\Q$ are abelian.\nClass field theory describes the abelian extensions of number fields and local fields.\n\n## Explanations\n\n### L1 human\n\nAn extension is abelian when its symmetries commute with each other.\nThen the order in which two symmetries are applied does not matter.\nFor example, $\\Q(\\sqrt{2}, \\sqrt{3})$ over $\\Q$ has four symmetries, each changing the signs of $\\sqrt{2}$ and of $\\sqrt{3}$ independently.\nChanging the two signs in either order gives the same result, so the extension is abelian.\n",
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