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<title>Class number problem</title>

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Class number problem

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of <a href="Imaginary_quadratic_field" class="mw-redirect" title="Imaginary quadratic field">imaginary quadratic fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ({\sqrt {d}})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {d}})}</annotation>

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</math><img src="./9757d116555dd605228d352039f5f491d2967c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.769ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {d}})}" loading="lazy"> (for negative integers d) having <a href="Class_number_(number_theory)" class="mw-redirect" title="Class number (number theory)">class number</a> n. It is named after <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>. It can also be stated in terms of <a href="Discriminant_of_an_algebraic_number_field" title="Discriminant of an algebraic number field">discriminants</a>. There are related questions for real quadratic fields and for the behavior as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\to -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle d\to -\infty }</annotation>

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</math><img src="./93509c2969d44c61de792a707b43e0e35e97d9cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.962ex; height:2.343ex;" alt="{\displaystyle d\to -\infty }" loading="lazy">.

The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.

<div class="mw-heading mw-heading2"><h2 id="Gauss's_original_conjectures">Gauss's original conjectures</h2></div>

The problems are posed in Gauss's <a href="Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a> of 1801 (Section V, Articles 303 and 304).<a href="#cite_note-1">[1]</a>

Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture.

<dl><dt>Gauss conjecture (class number tends to infinity)</dt>

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<annotation encoding="application/x-tex">{\displaystyle h(d)\to \infty {\text{ as }}d\to -\infty .}</annotation>

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</math><img src="./5c0d07e2aced83da2d8f8c6f990162449f082f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.151ex; height:2.843ex;" alt="{\displaystyle h(d)\to \infty {\text{ as }}d\to -\infty .}" loading="lazy"></dd>

<dt>Gauss class number problem (low class number lists)</dt>

<dd>For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete.</dd>

<dt>Infinitely many real quadratic fields with class number one</dt>

<dd>Gauss conjectures that there are infinitely many real quadratic fields with class number one.</dd></dl>

The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.

<div class="mw-heading mw-heading2"><h2 id="Status">Status</h2></div>

<dl><dt>Gauss conjecture</dt>

<dd>solved, Heilbronn, 1934.<a href="#cite_note-GaussClassNumber-2">[2]</a></dd>

<dt>Low class number lists</dt>

<dd>class number 1: solved, Baker (1966), Stark (1967), Heegner (1952).</dd>

<dd>Class number 2: solved, Baker (1971), Stark (1971)<a href="#cite_note-irelandrosen-3">[3]</a></dd>

<dd>Class number 3: solved, Oesterlé (1985)<a href="#cite_note-irelandrosen-3">[3]</a></dd>

<dd>Class numbers h up to 100: solved, Watkins 2004<a href="#cite_note-watkins-4">[4]</a></dd>

<dt>Infinitely many real quadratic fields with class number one</dt>

<div class="mw-heading mw-heading2"><h2 id="Lists_of_discriminants_of_class_number_1">Lists of discriminants of class number 1</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Heegner_number" title="Heegner number">Heegner number</a></div>

For imaginary quadratic number fields, the (fundamental) <a href="Imaginary_quadratic_field" class="mw-redirect" title="Imaginary quadratic field">discriminants</a> of class number 1 are:

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<annotation encoding="application/x-tex">{\displaystyle d=-3,-4,-7,-8,-11,-19,-43,-67,-163.}</annotation>

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</math><img src="./c3283a1b9bdaaa7ee83cf38860a28e931016ef4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.943ex; height:2.509ex;" alt="{\displaystyle d=-3,-4,-7,-8,-11,-19,-43,-67,-163.}" loading="lazy"></dd></dl>

The non-fundamental discriminants of class number 1 are:

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<annotation encoding="application/x-tex">{\displaystyle d=-12,-16,-27,-28.}</annotation>

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</math><img src="./85d18a26d8423d1bfbb753dace32b19b6284e9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.595ex; height:2.509ex;" alt="{\displaystyle d=-12,-16,-27,-28.}" loading="lazy"></dd></dl>

Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are:

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<annotation encoding="application/x-tex">{\displaystyle d=-4,-8,-12,-16,-28.}</annotation>

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</math><img src="./d38978ffb9037e4afead437d1ab36c8ab65be8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.437ex; height:2.509ex;" alt="{\displaystyle d=-4,-8,-12,-16,-28.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Modern_developments">Modern developments</h2></div>

In 1934, <a href="Hans_Heilbronn" title="Hans Heilbronn">Hans Heilbronn</a> proved the Gauss conjecture.<a href="#cite_note-GaussClassNumber-2">[2]</a> Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.

Also in 1934, Heilbronn and <a href="Edward_Linfoot" title="Edward Linfoot">Edward Linfoot</a> showed that there were at most 10<a href="#cite_note-HeilbronnLinfoot-5">[5]</a> imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).

The result was ineffective (see <a href="Effective_results_in_number_theory" title="Effective results in number theory">effective results in number theory</a>): it did not give bounds on the size of the remaining field.

In later developments, the case n = 1 was first discussed by <a href="Kurt_Heegner" title="Kurt Heegner">Kurt Heegner</a>, using <a href="Modular_form" title="Modular form">modular forms</a> and <a href="Modular_equation" title="Modular equation">modular equations</a> to show that no further such field could exist. This work was not initially accepted; only with later work of <a href="Harold_Stark" title="Harold Stark">Harold Stark</a> and <a href="Bryan_Birch" class="mw-redirect" title="Bryan Birch">Bryan Birch</a> (e.g. on the <a href="Stark%E2%80%93Heegner_theorem" title="Stark–Heegner theorem">Stark–Heegner theorem</a> and <a href="Heegner_number" title="Heegner number">Heegner number</a>) was the position clarified and Heegner's work understood. Practically simultaneously, <a href="Alan_Baker_(mathematician)" title="Alan Baker (mathematician)">Alan Baker</a> proved what we now know as <a href="Baker's_theorem" title="Baker's theorem">Baker's theorem</a> on <a href="Linear_forms_in_logarithms" class="mw-redirect" title="Linear forms in logarithms">linear forms in logarithms</a> of <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>, which resolved the problem by a completely different method. The case n = 2 was tackled shortly afterwards, at least in principle, as an application of Baker's work.<a href="#cite_note-Baker-6">[6]</a>

The complete list of imaginary quadratic fields with class number 1 is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {d}})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {d}})}</annotation>

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</math><img src="./d435339ec9508c7548b57d117f7566a91cda1b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.969ex; height:3.176ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {d}})}" loading="lazy"> where d is one of

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<annotation encoding="application/x-tex">{\displaystyle -1,-2,-3,-7,-11,-19,-43,-67,-163.}</annotation>

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</math><img src="./52901feee21ff6484467b78a8033dc2ce40c0d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.628ex; height:2.509ex;" alt="{\displaystyle -1,-2,-3,-7,-11,-19,-43,-67,-163.}" loading="lazy"></dd></dl>

The general case awaited the discovery of <a href="Dorian_Goldfeld" class="mw-redirect" title="Dorian Goldfeld">Dorian Goldfeld</a> in 1976 that the class number problem could be connected to the <a href="L-function" title="L-function">L-functions</a> of <a href="Elliptic_curve" title="Elliptic curve">elliptic curves</a>.<a href="#cite_note-Goldfeld-7">[7]</a> This effectively reduced the question of effective determination to one about establishing the existence of a multiple zero of such an L-function.<a href="#cite_note-Goldfeld-7">[7]</a> With the proof of the <a href="Gross%E2%80%93Zagier_theorem" class="mw-redirect" title="Gross–Zagier theorem">Gross–Zagier theorem</a> in 1986, a complete list of imaginary quadratic fields with a given class number could be specified by a finite calculation. All cases up to n = 100 were computed by Watkins in 2004.<a href="#cite_note-watkins-4">[4]</a> The class number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {-d}})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {-d}})}</annotation>

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</math><img src="./c4287173b611294985042ccdaf84e7e2a75c38b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.777ex; height:3.343ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {-d}})}" loading="lazy"> for d = 1, 2, 3, ... is

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<annotation encoding="application/x-tex">{\displaystyle 1,1,1,1,2,2,1,1,1,2,1,1,2,4,2,1,4,1,1,2,4,2,3,2,1,6,1,1,6,4,3,1,...}</annotation>

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</math><img src="./0c8a04f3851b4affa1347ad8b615405a2fab8b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:72.999ex; height:2.509ex;" alt="{\displaystyle 1,1,1,1,2,2,1,1,1,2,1,1,2,4,2,1,4,1,1,2,4,2,3,2,1,6,1,1,6,4,3,1,...}" loading="lazy"> (sequence <a href="https://oeis.org/A202084" class="extiw external" title="oeis:A202084">A202084</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Real_quadratic_fields">Real quadratic fields</h2></div>

The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but h log ε, where ε is a <a href="Fundamental_unit_(number_theory)" title="Fundamental unit (number theory)">fundamental unit</a>. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.

The Cohen–Lenstra heuristics<a href="#cite_note-FOOTNOTECohen1993ch._5.10-8">[8]</a> are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.45% of the fields obtained by adjoining the square root of a prime will have class number 1, a result that agrees with computations.<a href="#cite_note-9">[9]</a>

<ul><li><a href="List_of_number_fields_with_class_number_one" title="List of number fields with class number one">List of number fields with class number one</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFStark2007" class="citation book cs1"><a href="Harold_Stark" title="Harold Stark">Stark, H. M.</a> (2007). "The Gauss Class-Number Problems". In <a href="William_Duke_(mathematician)" title="William Duke (mathematician)">Duke, William</a>; <a href="Yuri_Tschinkel" title="Yuri Tschinkel">Tschinkel, Yuri</a> (eds.). <a rel="nofollow" class="external text" href="https://www.claymath.org/wp-content/uploads/2022/03/cmip07c.pdf">Analytic Number Theory: A Tribute to Gauss and Dirichlet</a> (pdf). Clay Mathematics Proceedings. Vol. 7. <a href="American_Mathematical_Society" title="American Mathematical Society">AMS</a> & <a href="Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a>. pp. 247–256. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4307-9</bdi>. Retrieved 2023-12-19.</cite>

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<li id="cite_note-GaussClassNumber-2">^ <a href="#cite_ref-GaussClassNumber_2-0">a</a> <a href="#cite_ref-GaussClassNumber_2-1">b</a> <cite id="CITEREFHeilbronn1934" class="citation journal cs1">Heilbronn, Hans (1934). <a rel="nofollow" class="external text" href="https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-5.1.150">"ON THE CLASS-NUMBER IN IMAGINARY QUADRATIC FIELDS"</a>. The Quarterly Journal of Mathematics. os-5 (1): 150–160. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2Fos-5.1.150">10.1093/qmath/os-5.1.150</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5606">0033-5606</a>. Retrieved 2025-04-21.</cite>

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<li id="cite_note-irelandrosen-3">^ <a href="#cite_ref-irelandrosen_3-0">a</a> <a href="#cite_ref-irelandrosen_3-1">b</a> <cite id="CITEREFIrelandRosen1993" class="citation cs2">Ireland, K.; Rosen, M. (1993), A Classical Introduction to Modern Number Theory, New York, New York: Springer-Verlag, pp. 358–361, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-97329-6</bdi></cite>

</li>

<li id="cite_note-watkins-4">^ <a href="#cite_ref-watkins_4-0">a</a> <a href="#cite_ref-watkins_4-1">b</a> <cite id="CITEREFWatkins2004" class="citation cs2">Watkins, M. (2004), <a rel="nofollow" class="external text" href="https://www.ams.org/mcom/2004-73-246/S0025-5718-03-01517-5/home.html">Class numbers of imaginary quadratic fields</a>, Mathematics of Computation, vol. 73, pp. 907–938, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-03-01517-5">10.1090/S0025-5718-03-01517-5</a></cite>

</li>

<li id="cite_note-HeilbronnLinfoot-5"><a href="#cite_ref-HeilbronnLinfoot_5-0">^</a> <cite id="CITEREFHeilbronnLinfoot1934" class="citation journal cs1">Heilbronn, H.; Linfoot, E. H. (1934). <a rel="nofollow" class="external text" href="https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-5.1.293">"ON THE IMAGINARY QUADRATIC CORPORA OF CLASS-NUMBER ONE"</a>. The Quarterly Journal of Mathematics. os-5 (1): 293–301. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2Fos-5.1.293">10.1093/qmath/os-5.1.293</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5606">0033-5606</a>. Retrieved 2025-04-21.</cite>

</li>

<li id="cite_note-Baker-6"><a href="#cite_ref-Baker_6-0">^</a> <a href="#CITEREFBaker1990">Baker (1990)</a>

</li>

<li id="cite_note-Goldfeld-7">^ <a href="#cite_ref-Goldfeld_7-0">a</a> <a href="#cite_ref-Goldfeld_7-1">b</a> <a href="#CITEREFGoldfeld1985">Goldfeld (1985)</a>

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<li id="cite_note-FOOTNOTECohen1993ch._5.10-8"><a href="#cite_ref-FOOTNOTECohen1993ch._5.10_8-0">^</a> <a href="#CITEREFCohen1993">Cohen 1993</a>, ch. 5.10.

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFte_RieleWilliams2003" class="citation journal cs1">te Riele, Herman; Williams, Hugh (2003). <a rel="nofollow" class="external text" href="http://www.emis.de/journals/EM/expmath/volumes/12/12.1/pp99_113.pdf">"New Computations Concerning the Cohen-Lenstra Heuristics"</a> (PDF). Experimental Mathematics. 12 (1): 99–113. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10586458.2003.10504715">10.1080/10586458.2003.10504715</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10221100">10221100</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFGoldfeld1985" class="citation cs2">Goldfeld, Dorian (July 1985), <a rel="nofollow" class="external text" href="https://www.ams.org/bull/1985-13-01/S0273-0979-1985-15352-2/S0273-0979-1985-15352-2.pdf">"Gauss' Class Number Problem For Imaginary Quadratic Fields"</a> (PDF), <a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>, 13 (1): 23–37, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1985-15352-2">10.1090/S0273-0979-1985-15352-2</a></cite></li>

<li><cite id="CITEREFHeegner1952" class="citation cs2"><a href="Kurt_Heegner" title="Kurt Heegner">Heegner, Kurt</a> (1952), "Diophantische Analysis und Modulfunktionen", <a href="Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a>, 56 (3): 227–253, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01174749">10.1007/BF01174749</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0053135">0053135</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120109035">120109035</a></cite></li>

<li><cite id="CITEREFCohen1993" class="citation cs2">Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-55640-4</bdi></cite></li>

<li><cite id="CITEREFBaker1990" class="citation cs2">Baker, Alan (1990), <a rel="nofollow" class="external text" href="https://books.google.com/books?isbn=052139791X">Transcendental number theory</a>, Cambridge Mathematical Library (2nd ed.), <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-39791-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0422171">0422171</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GausssClassNumberProblem.html">"Gauss's Class Number Problem"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></div><div><div class="zim-footer">

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