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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $G$ be a group that [[acts|0WCY42EV9V]] on $R$ by ring automorphisms, written $r \\mapsto g \\cdot r$.\nThe *ring of invariants* of the action is\n$$R^G = \\set{ r \\in R : g \\cdot r = r \\text{ for all } g \\in G }.$$\n\n## Remarks\n\n$R^G$ is a [[subring|0GNCRPYXYV]] of $R$, because each $g$ acts by a ring homomorphism and so fixes $0$, $1$, sums and products of fixed elements.\nThe main case is a group $G$ acting linearly on a finite-dimensional vector space $V$ over a field $k$.\nThen $G$ acts on the [[polynomial ring|0SDC934R52]] $k[V]$ of polynomial functions by $(g \\cdot f)(v) = f(g^{-1} v)$.\nEmmy Noether proved that for a finite group $G$ acting by $k$-algebra automorphisms on a finitely generated commutative $k$-algebra $R$, the ring $R^G$ is finitely generated.\n\nExample: the symmetric group $S_n$ acts on $k[x_1, \\ldots, x_n]$ by permuting the variables.\nIts ring of invariants is the ring of [[symmetric polynomials|0VHSHB9B6J]], which is a polynomial ring in the elementary symmetric polynomials by the [[fundamental theorem of symmetric polynomials|0GW1ZZH0R4]].\n\nExample: let $G = \\set{1, g}$ act on $k[x, y]$ by $g \\cdot x = -x$ and $g \\cdot y = -y$, where the characteristic of $k$ is not $2$.\nA monomial $x^a y^b$ is fixed exactly when $a + b$ is even, so $k[x, y]^G = k[x^2, xy, y^2]$.\n\nNon-example: in the last example, $x \\notin k[x, y]^G$, because $g \\cdot x = -x \\ne x$.\n",
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