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{{Short description|Listing all imaginary quadratic fields with a given class number}}
In [[mathematics]], the '''Gauss class number problem''' ('''for imaginary quadratic fields'''), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of [[imaginary quadratic field]]s <math>\mathbb{Q}(\sqrt{d})</math> (for negative integers ''d'') having [[class number (number theory)|class number]] ''n''. It is named after [[Carl Friedrich Gauss]]. It can also be stated in terms of [[Discriminant of an algebraic number field|discriminants]]. There are related questions for real quadratic fields and for the behavior as <math>d \to -\infty</math>.

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.

==Gauss's original conjectures==
The problems are posed in Gauss's [[Disquisitiones Arithmeticae]] of 1801 (Section V, Articles 303 and 304).<ref>{{Cite book |last=Stark |first=H. M. |url=https://www.claymath.org/wp-content/uploads/2022/03/cmip07c.pdf |title=Analytic Number Theory{{colon}} A Tribute to Gauss and Dirichlet |publisher=[[American Mathematical Society|AMS]] & [[Clay Mathematics Institute]] |year=2007 |isbn=978-0-8218-4307-9 |editor-last=Duke |editor-first=William |editor-link=William Duke (mathematician) |series=Clay Mathematics Proceedings |volume=7 |pages=247–256 |language=en |chapter=The Gauss Class-Number Problems |format=pdf |author-link=Harold Stark |access-date=2023-12-19 |editor-last2=Tschinkel |editor-first2=Yuri |editor-link2=Yuri Tschinkel}}</ref>

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.
;Gauss conjecture (class number tends to infinity): <math>h(d) \to \infty\text{ as }d\to -\infty.</math>
;Gauss class number problem (low class number lists): 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.
;Infinitely many real quadratic fields with class number one: Gauss conjectures that there are infinitely many real quadratic fields with class number one.

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.

==Status==
;Gauss conjecture: solved, Heilbronn, 1934.<ref name="GaussClassNumber">{{cite journal | last=Heilbronn | first=Hans | title=ON THE CLASS-NUMBER IN IMAGINARY QUADRATIC FIELDS | journal=The Quarterly Journal of Mathematics | volume=os-5 | issue=1 | date=1934 | issn=0033-5606 | doi=10.1093/qmath/os-5.1.150 | pages=150–160 | url=https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-5.1.150 | access-date=2025-04-21| url-access=subscription }}</ref>
;Low class number lists: class number 1: solved, Baker (1966), Stark (1967), Heegner (1952).
:Class number 2: solved, Baker (1971), Stark (1971)<ref name=irelandrosen>{{citation | last1 = Ireland | first1 = K. |last2 = Rosen | first2 = M. | title = A Classical Introduction to Modern Number Theory | publisher = Springer-Verlag | year = 1993 | location = New York, New York | pages = 358–361 | isbn = 978-0-387-97329-6}}</ref>
:Class number 3: solved, Oesterlé (1985)<ref name=irelandrosen/>
:Class numbers h up to 100: solved, Watkins 2004<ref name=watkins>{{citation | last1 = Watkins | first1 = M. | title = Class numbers of imaginary quadratic fields | series = Mathematics of Computation | volume = 73 | issue = 246 | year = 2004 | pages = 907–938 | url=https://www.ams.org/mcom/2004-73-246/S0025-5718-03-01517-5/home.html| doi = 10.1090/S0025-5718-03-01517-5 | doi-access =free }}</ref>
;Infinitely many real quadratic fields with class number one: Open.

==Lists of discriminants of class number 1==
{{details|Heegner number}}
For imaginary quadratic number fields, the (fundamental) [[Imaginary quadratic field#Discriminant|discriminants]] of class number 1 are:
:<math>d=-3,-4,-7,-8,-11,-19,-43,-67,-163.</math>
The non-fundamental discriminants of class number 1 are:
:<math>d=-12,-16,-27,-28.</math>
Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are:
:<math>d=-4,-8,-12,-16,-28.</math>

==Modern developments==
In 1934, [[Hans Heilbronn]] proved the Gauss conjecture.<ref name="GaussClassNumber" /> Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.

Also in 1934, Heilbronn and [[Edward Linfoot]] showed that there were at most 10<ref name="HeilbronnLinfoot">{{cite journal | last=Heilbronn | first=H. | last2=Linfoot | first2=E. H. | title=ON THE IMAGINARY QUADRATIC CORPORA OF CLASS-NUMBER ONE | journal=The Quarterly Journal of Mathematics | volume=os-5 | issue=1 | date=1934 | issn=0033-5606 | doi=10.1093/qmath/os-5.1.293 | pages=293–301 | url=https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-5.1.293 | access-date=2025-04-21| url-access=subscription }}</ref> imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).
The result was ineffective (see [[effective results in number theory]]): it did not give bounds on the size of the remaining field.

In later developments, the case ''n'' = 1 was first discussed by [[Kurt Heegner]], using [[modular form]]s and [[modular equation]]s to show that no further such field could exist. This work was not initially accepted; only with later work of [[Harold Stark]] and [[Bryan Birch]] (e.g. on the [[Stark–Heegner theorem]] and [[Heegner number]]) was the position clarified and Heegner's work understood. Practically simultaneously, [[Alan Baker (mathematician)|Alan Baker]] proved what we now know as [[Baker's theorem]] on [[linear forms in logarithms]] of [[algebraic number]]s, 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.<ref name=Baker>{{harvtxt|Baker|1990}}</ref>

The complete list of imaginary quadratic fields with class number 1 is <math>\mathbf{Q}(\sqrt{d})</math> where ''d'' is one of
:<math>-1, -2, -3, -7, -11, -19, -43, -67, -163.</math>

The general case awaited the discovery of [[Dorian Goldfeld]] in 1976 that the class number problem could be connected to the [[L-function|''L''-function]]s of [[elliptic curve]]s.<ref name=Goldfeld>{{harvtxt|Goldfeld|1985}}</ref> This effectively reduced the question of effective determination to one about establishing the existence of a multiple zero of such an ''L''-function.<ref name=Goldfeld/> With the proof of the [[Gross–Zagier theorem]] 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.<ref name=watkins/> The class number of <math>\mathbf{Q}(\sqrt{-d})</math> for ''d'' = 1, 2, 3, ... is

:<math>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, ...</math> {{OEIS|A202084}}.

==Real quadratic fields==
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 [[fundamental unit (number theory)|fundamental unit]]. 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{{sfn|Cohen|1993|loc=ch. 5.10}} 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.<ref>{{Cite journal
| last1 = te Riele
| first1 = Herman
| last2 = Williams
| first2 = Hugh
| year = 2003
| title = New Computations Concerning the Cohen-Lenstra Heuristics
| journal = Experimental Mathematics
| volume = 12
| issue = 1
| pages = 99–113
| url = http://www.emis.de/journals/EM/expmath/volumes/12/12.1/pp99_113.pdf
| doi=10.1080/10586458.2003.10504715| s2cid = 10221100
}}</ref>

==See also==
* [[List of number fields with class number one]]

==Notes==
{{Reflist}}

==References==
* {{Citation
| last1=Baker
| first1=Alan
| title=Transcendental number theory
| url=https://books.google.com/books?isbn=052139791X
| publisher=[[Cambridge University Press]]
| edition=2nd | series=Cambridge Mathematical Library
| isbn=978-0-521-39791-9
| mr=0422171
| year=1990}}
* {{citation
| last1 = Cohen | first1 = Henri
| title = A Course in Computational Algebraic Number Theory
| publisher = [[Springer Science+Business Media|Springer]]
| location = Berlin
| year = 1993
| isbn = 978-3-540-55640-4}}
* {{Citation
| last = Goldfeld
| first = Dorian
|date=July 1985
| title = Gauss' Class Number Problem For Imaginary Quadratic Fields
| journal = [[Bulletin of the American Mathematical Society]]
| volume = 13
| issue = 1
| pages = 23–37
| url = https://www.ams.org/bull/1985-13-01/S0273-0979-1985-15352-2/S0273-0979-1985-15352-2.pdf
| doi = 10.1090/S0273-0979-1985-15352-2| doi-access = free
}}
* {{citation | last=Heegner | first=Kurt | authorlink=Kurt Heegner | doi=10.1007/BF01174749|mr=0053135 | title=Diophantische Analysis und Modulfunktionen | journal=[[Mathematische Zeitschrift]] | volume=56 | issue=3 | year=1952 | pages=227–253| s2cid=120109035 }}

==External links==
* {{MathWorld|title=Gauss's Class Number Problem|urlname=GausssClassNumberProblem}}

[[Category:Algebraic number theory]]
[[Category:Mathematical problems]]
[[Category:Unsolved problems in number theory]]

Class number problem - Revision history

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