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  "body": "\n## Definition\n\nLet $L/K$ be a [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $\\Gal(L/K)$.\nThe extension $L/K$ is *abelian* if $\\Gal(L/K)$ is an [[abelian group|071DDFVG4J]].\n\n## Remarks\n\nExample: $\\Q(\\sqrt{2}, \\sqrt{3})/\\Q$ is Galois of degree $4$.\nIts Galois group consists of the four maps $\\sqrt{2} \\mapsto \\pm\\sqrt{2}$, $\\sqrt{3} \\mapsto \\pm\\sqrt{3}$, which commute, so the extension is abelian.\nNon-example: the splitting field $L = \\Q(\\sqrt[3]{2}, \\omega)$ of $x^3 - 2$, with $\\omega = e^{2\\pi i/3}$, is Galois over $\\Q$.\nIts Galois group permutes the three roots of $x^3 - 2$ in all six ways, so it is isomorphic to $S_3$, which is not abelian.\nEvery subextension of a finite abelian extension is abelian, because every subgroup of an abelian group is normal and every quotient is abelian.\n",
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