Class Number Problem For Imaginary Quadratic Fields
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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 In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
s \mathbb(\sqrt) (for negative integers ''d'') having class number ''n''. It is named after
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as d \to -\infty. 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 The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
of 1801 (Section V, Articles 303 and 304).The Gauss Class-Number Problems
by H. M. Stark 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): h(d) \to \infty\textd\to -\infty. ;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. ;Low class number lists: Class number 1: solved, Baker (1966), Stark (1967), Heegner (1952). :Class number 2: solved, Baker (1971), Stark (1971) :Class number 3: solved, Oesterlé (1985) :Class numbers h up to 100: solved, Watkins 2004 ;Infinitely many real quadratic fields with class number one: Open.


Lists of discriminants of class number 1

For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are: :d=-3,-4,-7,-8,-11,-19,-43,-67,-163. The non-fundamental discriminants of class number 1 are: :d=-12,-16,-27,-28. Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are: :d=-4,-8,-12,-16,-28.


Modern developments

In 1934,
Hans Heilbronn Hans Arnold Heilbronn (8 October 1908 – 28 April 1975) was a mathematician. Education He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised ...
proved the Gauss Conjecture. 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 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 For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where ...
): 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 Kurt Heegner (; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–He ...
, using modular forms and modular equations to show that no further such field could exist. This work was not initially accepted; only with later work of
Harold Stark Harold Mead Stark (born August 6, 1939 in Los Angeles, California) is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earl ...
and
Bryan Birch Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture. Biography Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...
(e.g. on the Stark–Heegner theorem and Heegner number) was the position clarified and Heegner's work understood. Practically simultaneously, Alan Baker proved what we now know as
Baker's theorem In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendent ...
on
linear forms in logarithms In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendenta ...
of algebraic numbers, 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. The complete list of imaginary quadratic fields with class number one is \mathbf(\sqrt) with ''k'' one of :-1, -2, -3, -7, -11, -19, -43, -67, -163. The general case awaited the discovery of
Dorian Goldfeld Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University. Professional career Goldfeld received his B.S. degree in 1967 from Columbia University. ...
in 1976 that the class number problem could be connected to the
L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ri ...
s of
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s. This effectively reduced the question of effective determination to one about establishing the existence of a multiple zero of such an L-function. 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. A full list of class numbers


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. 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> 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.


See also

* List of number fields with class number one


Notes


References

* * * *


External links

* {{MathWorld, title=Gauss's Class Number Problem, urlname=GausssClassNumberProblem Algebraic number theory Mathematical problems Unsolved problems in number theory