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  "body": "\n## Summary\n\nThe ring of invariants of a group acting on a commutative ring is the subring of elements fixed by every element of the group.\nFor a group acting linearly on a vector space, it is the ring of polynomial functions that do not change under the group.\nIt is the central object of invariant theory.\nWhether it is finitely generated is the subject of [[Hilbert's fourteenth problem|M5N08HD75V]].\nIts ring-theoretic properties are the subject of results such as the [[Hochster–Roberts theorem|56107GMFYV]].\nGeometrically, it is the coordinate ring of the quotient of the space by the group.\n\n## Explanations\n\n### L1 human\n\nLet the symmetric group permute the variables $x_1, \\ldots, x_n$.\nThe polynomials that do not change, such as $x_1 + \\cdots + x_n$ and $x_1 x_2 \\cdots x_n$, are the symmetric polynomials, and they form a ring.\nAnother example: over the rational numbers, the map $(x, y) \\mapsto (-x, -y)$ fixes exactly the polynomials whose terms all have even degree.\nThat ring is generated by $x^2$, $xy$ and $y^2$, which satisfy the relation $x^2 \\cdot y^2 = (xy)^2$.\nRings of invariants describe the \"shape\" of a space after points in the same orbit are glued together.\n",
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