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  "body": "\n## Statement\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n \\ge 2$ with [[discriminant|E0VZTNPJXA]] $d_K$.\nLet $r_2$ be the number of pairs of complex conjugate embeddings of $K$.\n\n1. (Minkowski) The discriminant satisfies\n$$\\abs{d_K} \\ge \\left(\\frac{\\pi}{4}\\right)^{2 r_2} \\left(\\frac{n^n}{n!}\\right)^2 \u003e 1.$$\nIn particular $\\abs{d_K} \\ge 3$, and some prime number [[ramifies|H5RC67SD24]] in $K$.\n2. (Hermite) For each real number $D$, there are only finitely many number fields $K \\subset \\C$ with $\\abs{d_K} \\le D$.\n\n## Remarks\n\nPart 1 follows from the [[Minkowski bound|1TQ2F9CFVE]]: every ideal class contains a nonzero integral ideal of norm at least $1$ and at most $M_K$, so $M_K \\ge 1$.\nSince $2 r_2 \\le n$ and $\\pi/4 \u003c 1$, the right side is at least $(\\pi/4)^n (n^n/n!)^2$, which is $\\pi^2/4 \u003e 2$ for $n = 2$ and increases with $n$.\nA prime $p$ ramifies in $K$ exactly when $p$ divides $d_K$, so $\\abs{d_K} \u003e 1$ gives a ramified prime.\n\nFor part 2, the bound of part 1 first bounds the degree $n$.\nThen [[Minkowski's theorem|0K387S2PKQ]] on convex bodies gives a primitive element of $K$ whose conjugates are bounded in terms of $D$.\nSo its minimal polynomial has bounded integer coefficients, and there are finitely many such polynomials.\n\nExample: $\\Q(\\sqrt{-3})$ has $\\abs{d_K} = 3$, so the bound $\\abs{d_K} \\ge 3$ for $n = 2$ is sharp.\n\n## Explanations\n\n### L1 human\n\nEvery number field other than $\\Q$ has a discriminant of absolute value larger than $1$.\nSo in every proper extension of $\\Q$ some prime number breaks into repeated factors.\nAlso, only finitely many number fields have a discriminant below a given size.\n\n### L2 human\n\nPart 1 says that $\\Q$ has no nontrivial extension that is unramified at all primes.\nMany other number fields do have such extensions, and the abelian ones are described by the [[Hilbert class field|WWCJ7CR2R3]].\nPart 2 makes tables of number fields ordered by discriminant possible, since each range contains finitely many fields.\n",
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