Gauss conjectured in 1801 that there are infinitely many real quadratic fields with ideal class number one. Despite supporting heuristics (Cohen–Lenstra predicts a positive proportion, about 75.446% for prime discriminants) and extensive numerical evidence, no unconditional proof or disproof is known: all unconditional results determine the class-number-one fields only within explicit one- and two-parameter families, each containing finitely many of them. The problem asks for a proof or refutation, in full generality, of the statement that infinitely many squarefree positive integers dd satisfy h(Q(d))=1h(\mathbb{Q}(\sqrt{d}))=1.

Background

A quadratic field is K=Q(d)K=\mathbb{Q}(\sqrt{d}) for a squarefree integer dd; it is called real when d>0d>0 and imaginary when d<0d<0. Its ring of integers OK\mathcal{O}_K need not have unique factorization; the failure is measured by the ideal class group, the group of nonzero fractional ideals of OK\mathcal{O}_K modulo principal ideals, which is finite, and its order h(K)h(K) is the class number. Thus h(K)=1h(K)=1 exactly when every ideal of OK\mathcal{O}_K is principal, i.e. OK\mathcal{O}_K is a unique factorization domain.

In the Disquisitiones Arithmeticae (1801), working with the equivalent language of binary quadratic forms, Gauss studied class numbers of quadratic fields. The imaginary side is completely understood: Gauss's list of the nine imaginary quadratic fields of class number one (d∈{−1,−2,−3,−7,−11,−19,−43,−67,−163}d\in\{-1,-2,-3,-7,-11,-19,-43,-67,-163\}) was proved complete by Stark in 1967, and Goldfeld's survey recounts how the full imaginary class number problem was subsequently made effective. For real quadratic fields the situation is fundamentally harder. By Dirichlet's unit theorem the unit group of a real quadratic field is infinite, generated by a fundamental unit whose logarithm is the regulator R(d)R(d); the analytic class number formula relates the class number to the central value of a Dirichlet LL-function (the Dirichlet series attached to the quadratic character χd\chi_d, L(s,χd)=∑n≥1χd(n)n−sL(s,\chi_d)=\sum_{n\ge1}\chi_d(n)n^{-s}) via h(d)R(d)=d L(1,χd)/2h(d)R(d)=\sqrt{d}\,L(1,\chi_d)/2 for d>0d>0 squarefree with d≡1(mod4)d\equiv1\pmod4 (with the usual modification otherwise). The regulator fluctuates erratically with dd and no known technique controls it over infinitely many dd, which is the essential obstruction that made the imaginary methods fail here.

Two kinds of partial information exist. First, genus theory: the 2-rank of the class group equals one less than the number of ramified primes, so any real quadratic field with prime discriminant has odd class number, and Dirichlet's theorem on primes then gives infinitely many real quadratic fields with odd class number — but nothing toward class number exactly one. Second, Cohen and Lenstra (1984) proposed heuristics modeling the class group of a real quadratic field by a random finite abelian group weighted by 1/∣Aut∣1/|\mathrm{Aut}|; they predict that a positive proportion of real quadratic fields have class number one (about 75.446% of prime discriminants), a prediction strongly supported by large-scale computations such as those of te Riele and Williams. However, these heuristics remain unproven, and class number one is a pointwise statement about individual fields, beyond what average-value techniques can reach.

Unconditional theorems are confined to Richaud–Degert type discriminants d=(an)2+kad=(an)^2+ka, k∈{1,2,4}k\in\{1,2,4\}, families in which the continued-fraction expansion of d\sqrt{d} makes the fundamental unit explicit. Biró (2003) resolved Yokoi's conjecture and Chowla's conjecture, determining exactly the class-number-one fields in such families; Lapkova (2012), Biró and Lapkova (2015), Hoque and Kotyada (2021), and Biró (2022) settled further families of the same shape. Each such family contains only finitely many class-number-one fields (ineffectively, by Siegel's theorem on the growth of L(1,χ)L(1,\chi)), so these methods cannot produce infinitude. Feaver's classification of imaginary multiquadratic fields of class number one recalls Gauss's real-quadratic conjecture in its introduction precisely as the still-open counterpart of the imaginary problem.

Problem Statement

Gauss's conjecture (1801), in modern terms: prove or disprove that there are infinitely many squarefree positive integers dd such that the real quadratic field Q(d)\mathbb{Q}(\sqrt{d}) has ideal class number one,

∣{ d∈Z>0:d squarefree, h(Q(d))=1 }∣=∞.\left|\{\,d\in\mathbb{Z}_{>0}: d\text{ squarefree},\ h(\mathbb{Q}(\sqrt{d}))=1\,\}\right|=\infty.

The question is to be answered in full generality, with no restriction on dd (no bound, congruence, or family shape) and no unproved assumption such as the generalized Riemann hypothesis (GRH, the extension of the Riemann hypothesis to Dirichlet and other LL-functions) or the Cohen–Lenstra heuristics.

References

  1. Amy Feaver, Imaginary Multiquadratic Fields of Class Number 1, Journal of Number Theory 174 (2017) 93–117. DOI: 10.1016/j.jnt.2016.10.010, https://doi.org/10.1016/j.jnt.2016.10.010 (arXiv:1608.04419, https://arxiv.org/abs/1608.04419)
  2. Carl Friedrich Gauss, Disquisitiones Arithmeticae (English edition), Springer, 1986. ISBN 978-0-387-96254-2, DOI: 10.1007/978-1-4939-7560-0, https://doi.org/10.1007/978-1-4939-7560-0
  3. H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Mathematical Journal 14 (1967) 1–27. DOI: 10.1307/mmj/1028999653, https://doi.org/10.1307/mmj/1028999653
  4. Dorian Goldfeld, Gauss' class number problem for imaginary quadratic fields, Bulletin of the American Mathematical Society 13 (1985) 23–37. DOI: 10.1090/s0273-0979-1985-15352-2, https://doi.org/10.1090/s0273-0979-1985-15352-2
  5. H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, in: Number Theory Noordwijkerhout 1983, Lecture Notes in Mathematics 1068, Springer, 1984, pp. 33–62. DOI: 10.1007/BFb0099440, https://doi.org/10.1007/BFb0099440
  6. Herman te Riele and Hugh Williams, New Computations Concerning the Cohen-Lenstra Heuristics, Experimental Mathematics 12 (2003) 99–113. DOI: 10.1080/10586458.2003.10504715, https://doi.org/10.1080/10586458.2003.10504715
  7. András Biró, Yokoi's conjecture, Acta Arithmetica 106 (2003) 85–104. DOI: 10.4064/aa106-1-6, https://doi.org/10.4064/aa106-1-6
  8. András Biró, Chowla's conjecture, Acta Arithmetica 107 (2003) 179–194. DOI: 10.4064/aa107-2-5, https://doi.org/10.4064/aa107-2-5
  9. Kostadinka Lapkova, Class number one problem for real quadratic fields of a certain type, Acta Arithmetica 153 (2012) 281–298. DOI: 10.4064/aa153-3-4, https://doi.org/10.4064/aa153-3-4
  10. András Biró and Kostadinka Lapkova, The class number one problem for the real quadratic fields Q((an)2+4a)\mathbb{Q}(\sqrt{(an)^2+4a}), arXiv:1508.05644 (2015). https://arxiv.org/abs/1508.05644
  11. Azizul Hoque and Srinivas Kotyada, Class number one problem for the real quadratic fields Q(m2+2r)\mathbb{Q}(\sqrt{m^2+2r}), Archiv der Mathematik 116 (2021) 33–36. DOI: 10.1007/s00013-020-01520-w, https://doi.org/10.1007/s00013-020-01520-w
  12. András Biró, Class number one problem for a family of real quadratic fields, Mathematika 68 (2022) 1221–1232. DOI: 10.1112/mtk.12164, https://doi.org/10.1112/mtk.12164