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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]].\nThen the [[ideal class group|WQ46BD3ARY]] $\\operatorname{Cl}(K)$ is finite.\n\n## Remarks\n\nThe standard proof uses the [[Minkowski bound|1TQ2F9CFVE]].\nLet $n = [K : \\Q]$, let $2r_2$ be the number of non-real embeddings of $K$ into $\\C$, and let $d_K$ be the [[discriminant|E0VZTNPJXA]] of $K$.\nBy [[Minkowski's theorem|0K387S2PKQ]] on lattice points, every ideal class contains an integral ideal $\\mathfrak{a}$ with [[norm|M54ADM0CVF]]\n$$N(\\mathfrak{a}) \\le \\frac{n!}{n^n} \\left(\\frac{4}{\\pi}\\right)^{r_2} \\sqrt{\\abs{d_K}}.$$\nThere are only finitely many ideals of $\\mathcal{O}_K$ of bounded norm, so there are finitely many classes.\nExample: for $K = \\Q(\\sqrt{-5})$, $n = 2$, $r_2 = 1$ and $d_K = -20$, so the bound is $\\tfrac12 \\cdot \\tfrac4\\pi \\cdot \\sqrt{20} \\approx 2.85$; the ideals of norm at most $2$ are $\\mathcal{O}_K$ and $(2, 1 + \\sqrt{-5})$, which gives $h_K \\le 2$.\nThe hypothesis that $K$ is a number field is needed: for a general [[Dedekind domain|7NPY2CN5KB]] the class group can be infinite.\n\n## Explanations\n\n### L1 human\n\nIn a number ring, unique factorisation can fail, and the class group measures by how much.\nThis theorem says that the failure is always finite: there are only finitely many kinds of ideals up to multiplication by numbers.\nIt turns the question of unique factorisation in a given field into a finite computation.\n\n### L2 human\n\nThe theorem is one of the two basic finiteness results of algebraic number theory, the other being [[Dirichlet's unit theorem|1D99CSKX6N]].\nBoth come from the geometry of numbers: $\\mathcal{O}_K$ embeds as a lattice in $\\R^n$, and Minkowski's convex body theorem produces small elements.\nThe explicit bound makes class groups computable; for example it shows at once that $\\Q(i)$, $\\Q(\\sqrt{-2})$ and $\\Q(\\sqrt{-3})$ have class number $1$.\nThe size of $h_K$ is described by the [[class number formula|55XX5A7XZV]], which links it to the residue of the [[Dedekind zeta function|KXS5DQDFYD]] at $s = 1$.\n",
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