In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, the Baker–Heegner–Stark theorem states precisely which
quadratic imaginary number fields admit
unique factorisation
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
in their
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
. It solves a special case of Gauss's
class number problem
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 fields \mathbb(\sqrt) (for negative integers ''d'') having ...
of determining the number of imaginary quadratic fields that have a given fixed
class number.
Let Q denote the set of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, and let ''d'' be a non-square
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. Then
Q() is a
finite extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — ...
of Q of degree 2, called a quadratic extension. The
class number of Q() is the number of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
s of the ring of integers of Q(), where two ideals ''I'' and ''J'' are equivalent
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
there exist
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
s (''a'') and (''b'') such that (''a'')''I'' = (''b'')''J''. Thus, the ring of integers of Q() is a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
(and hence a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
) if and only if the class number of Q() is equal to 1. The Baker–Heegner–Stark theorem can then be stated as follows:
:If ''d'' < 0, then the class number of Q() is equal to 1 if and only if
::
These are known as the
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriz ...
s.
This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:
:
where ''D'' is interpreted as the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
(either of the
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
or of an
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 ...
with
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
). This is more standard, as the ''D'' are then
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integ ...
s.
History
This result was first conjectured by
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 ...
in Section 303 of his ''
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 ...
'' (1798). It was essentially proven 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– ...
in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until
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 ear ...
gave a complete proof in 1967, which had many commonalities to Heegner's work, but sufficiently many differences that Stark considers the proofs to be different. Heegner "died before anyone really understood what he had done". Stark formally filled in the gap in Heegner's proof in 1969 (other contemporary papers produced various similar proofs by modular functions, but Stark concentrated on explicitly filling Heegner's gap).
Alan Baker gave a completely different proof slightly earlier (1966) than Stark's work (or more precisely Baker reduced the result to a finite amount of computation, with Stark's work in his 1963/4 thesis already providing this computation), and won the
Fields Medal for his methods. Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to only 2 logarithms, when the result was already known from 1949 by Gelfond and Linnik.
Stark's 1969 paper also cited the 1895 text by
Heinrich Martin Weber
Heinrich Martin Weber (5 March 1842, Heidelberg, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He is ...
and noted that if Weber had "only made the observation that the reducibility of
certain equationwould lead to a
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to ...
, the class-number one problem would have been solved 60 years ago".
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 ...
notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's ''Algebra'' to appreciate Heegner's achievement."
Deuring, Siegel, and Chowla all gave slightly variant proofs by
modular functions in the immediate years after Stark. Other versions in this genre have also cropped up over the years. For instance, in 1985, Monsur Kenku gave a proof using the
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact space, compact Riemann surface of genus (mathematics), genus with the highest possible order automorphism group for this genus, namely order orientation-preservi ...
(though again utilizing modular functions). And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).
The work of Gross and Zagier (1986) combined with that of Goldfeld (1976) also gives an alternative proof.
Real case
On the other hand, it is unknown whether there are infinitely many ''d'' > 0 for which Q() has class number 1. Computational results indicate that there are many such fields.
Number Fields with class number one provides a list of some of these.
Notes
References
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{{DEFAULTSORT:Stark-Heegner theorem
Theorems in algebraic number theory