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  "body": "\n## Problem\n\nIs every finite group $G$ isomorphic to the [[Galois group|0XVR72M51M]] $\\Gal(L/\\Q)$ of some [[Galois extension|0DS2F7GK05]] $L$ of the rational numbers $\\Q$?\n\n## Remarks\n\nThe problem is open.\nEvery finite abelian group is a Galois group over $\\Q$, realised inside a [[cyclotomic extension|VD6VC3ZMMY]] $\\Q(\\zeta_n)$.\nThe symmetric groups $S_n$ are Galois groups over $\\Q$; one route uses [[Hilbert's irreducibility theorem|N1V3YEYHNP]].\nEvery finite solvable group is a Galois group over $\\Q$, by [[Shafarevich's theorem|SDV692Z915]].\nThe same question can be asked over other fields; over $\\C(t)$ every finite group is a Galois group.\nOver a finite field $\\F_q$ only cyclic groups occur, since every finite extension of $\\F_q$ is cyclic.\n\n## Explanations\n\n### L1 human\n\nGalois theory attaches a finite group to each polynomial with rational coefficients.\nThe inverse problem asks whether every finite group appears in this way.\nMany families of groups are known to appear, but no proof covers all finite groups.\n",
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