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  "body": "\n## Problem\n\nFor each positive integer $h$, determine the complete list of imaginary [[quadratic fields|7SV4WA92SM]] $\\Q(\\sqrt{-d})$, with $d$ a positive squarefree integer, whose [[class number|KTKK9B1X41]] equals $h$.\n\n## Remarks\n\nThe problem goes back to Gauss, who stated it in the language of [[binary quadratic forms|SBDSYA037Q]].\nIt has three parts: show that the class number tends to infinity as $d \\to \\infty$, so that each list is finite; make this effective; and compute the lists.\n\nStatus:\nthe class number of $\\Q(\\sqrt{-d})$ tends to infinity with $d$ (Heilbronn), but this proof gives no effective bound.\nThe case $h = 1$ is the [[Stark–Heegner theorem|KZTFH832H0]], with nine fields, and the case $h = 2$ was solved by Baker and by Stark.\nGoldfeld, combined with the theorem of Gross and Zagier, gave an effective lower bound for the class number, which reduces each $h$ to a finite computation.\nWatkins completed the lists for all $h \\le 100$.\n\nThe analogue for real quadratic fields is open: it is not known whether infinitely many real quadratic fields have class number $1$.\n\n## Explanations\n\n### L1 human\n\nThe class number of a number field measures how badly unique factorisation fails in its integers.\nThe problem asks, for each value of the class number, which imaginary quadratic fields have it.\nFor class number $1$ the answer is nine fields; for larger class numbers the lists are known up to $100$ by computer-assisted proofs.\n",
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      "first_stated": "1801",
      "posed_by": "Carl Friedrich Gauss"
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