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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ be nonzero.\nLet $L$ be a [[splitting field|02NZZ1J0FH]] of $f$ over $K$.\nThe Galois group of $f$ over $K$ is the group $\\operatorname{Aut}(L/K)$ of the [[automorphisms of $L/K$|0XVR72M51M]].\n\n## Remarks\n\nBy the [[uniqueness of splitting fields|069A3BC3EP]], the group does not depend on the choice of $L$, up to isomorphism.\nEach automorphism permutes the distinct roots $r_1, \\dots, r_m$ of $f$ in $L$, and it is determined by this permutation, because the roots generate $L$. So the group is isomorphic to a subgroup of the symmetric group $S_m$.\nThe Galois group of $x^2 - 2$ over $\\mathbb{Q}$ has order $2$. The Galois group of $x^3 - 2$ over $\\mathbb{Q}$ is isomorphic to $S_3$.\nIf $f$ is separable, then $L/K$ is a [[Galois extension|0DS2F7GK05]].\n\n## Explanations\n\n### L1 human\n\nThe Galois group of a polynomial lists the ways to permute its roots that respect all the algebraic relations between them.\nIt measures how complicated the polynomial is.\nFor example, a polynomial over $\\mathbb{Q}$ can be solved by radicals exactly when its Galois group is solvable.\n",
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