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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n$, with ring of integers $\\mathcal{O}_K$ and [[discriminant|E0VZTNPJXA]] $d_K$.\nLet $r_2$ be the number of pairs of complex conjugate embeddings $K \\to \\C$.\nPut\n$$M_K = \\sqrt{\\abs{d_K}} \\left(\\frac{4}{\\pi}\\right)^{r_2} \\frac{n!}{n^n}.$$\nThen every class of the [[ideal class group|WQ46BD3ARY]] of $K$ contains a nonzero ideal $\\mathfrak{a} \\subseteq \\mathcal{O}_K$ whose [[norm|M54ADM0CVF]] satisfies $N(\\mathfrak{a}) \\le M_K$.\n\n## Remarks\n\nThe proof embeds $\\mathcal{O}_K$ and its ideals as [[lattices|WN9V7WSJKF]] in $\\R^n$.\n[[Minkowski's theorem|0K387S2PKQ]] on convex bodies then gives, in each nonzero ideal $\\mathfrak{b}$, a nonzero element $\\alpha$ with $\\abs{N_{K/\\Q}(\\alpha)} \\le M_K N(\\mathfrak{b})$.\nApplied to an ideal $\\mathfrak{b}$ in the inverse class, the ideal $\\alpha \\mathfrak{b}^{-1}$ is integral, lies in the given class and has norm at most $M_K$.\n\nConsequence: the class group is generated by the classes of the prime ideals of norm at most $M_K$.\nSince there are only finitely many ideals of bounded norm, the class group is finite.\n\nExample: for $K = \\Q(\\sqrt{-5})$ one has $n = 2$, $r_2 = 1$, $d_K = -20$, so $M_K = \\sqrt{20} \\cdot \\frac{4}{\\pi} \\cdot \\frac{2}{4} \\approx 2.85$.\nOnly the prime ideal $\\mathfrak{p} = (2, 1 + \\sqrt{-5})$ of norm $2$ needs checking.\nIt is not principal, since $a^2 + 5b^2 = 2$ has no integer solution, and $\\mathfrak{p}^2 = (2)$; so the class number is $2$.\nFor $K = \\Q(i)$, $M_K = 2 \\cdot \\frac{4}{\\pi} \\cdot \\frac{2}{4} \\approx 1.27 \u003c 2$, so every class contains the ideal $(1)$ and $\\Z[i]$ is a principal ideal domain.\n\n## Explanations\n\n### L1 human\n\nThe class group of a number field measures how far its integers are from having unique factorisation.\nThe Minkowski bound is an explicit number such that only the prime ideals below it matter.\nSo the class group of a given field can be computed by checking finitely many small primes.\n\n### L2 human\n\nThe bound is the standard tool for computing [[class numbers|KTKK9B1X41]] by hand for fields of small degree.\nSince every class contains an integral ideal of norm at least $1$, it also gives $M_K \\ge 1$, which yields a lower bound for $\\abs{d_K}$; this is the [[Hermite–Minkowski theorem|KMVW45B0TT]].\nThe same geometry of numbers proves [[Dirichlet's unit theorem|1D99CSKX6N]].\n",
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