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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] and $\\operatorname{Cl}(K)$ its [[ideal class group|WQ46BD3ARY]].\nThe *class number* of $K$ is the [[order|0B85S2V2DR]] of this group:\n$$h_K = \\#\\operatorname{Cl}(K).$$\nIt is a positive integer by the [[finiteness of the class number|F99RM1T02C]].\n\n## Remarks\n\nExample: for $K = \\Q(i)$, the ring of integers $\\Z[i]$ is a [[Euclidean domain|0PN13BB7BT]], hence a principal ideal domain, so $\\operatorname{Cl}(K)$ is trivial and $h_K = 1$.\nExample: for $K = \\Q(\\sqrt{-5})$, $\\operatorname{Cl}(K) \\cong \\Z/2\\Z$, so $h_K = 2$.\nNon-example: for a general [[Dedekind domain|7NPY2CN5KB]] the class group can be infinite, and then there is no finite class number.\nFor instance, let $E$ be the elliptic curve $y^2 = x^3 - x$ over $\\C$.\nThe class group of its affine coordinate ring $\\C[x, y]/(y^2 - x^3 + x)$ is isomorphic to the infinite group $E(\\C)$.\nBy the [[Stark–Heegner theorem|KZTFH832H0]], the imaginary quadratic fields $\\Q(\\sqrt{d})$ with $h = 1$ are those with $d = -1, -2, -3, -7, -11, -19, -43, -67, -163$.\n",
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