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Class Number 1 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 class number ''n''. It is named after Carl Friedrich Gauss. 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 of 1801 (Section V, Articles 303 and 304). are a set of more precise conjectures about the structure of class groups of quadratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Baker (mathematician)
Alan Baker (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Life Alan Baker was born in London on 19 August 1939. He attended Stratford Grammar School, East London, and his academic career started as a student of Harold Davenport, at University College London and later at Trinity College, Cambridge, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, transcendence, logarithmic forms, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift .
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Helmut Wielandt, and Olivier Debarre Olivier Debarre (born 1959) is a French mathematician who specializes in complex algebraic geometry.Debarr ... External links * * Mathematics journals Publications established in 1918 {{math ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Number Fields With Class Number One
This is an incomplete list of number fields with class number 1. It is believed that there are infinitely many such number fields, but this has not been proven. Definition The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1. Quadratic number fields These are of the form ''K'' = Q(), for a square-free integer ''d''. Real quadratic fields ''K'' is called real quadratic if ''d'' > 0. ''K'' has class number 1 for the following values of ''d'' : * 2*, 3, 5*, 6, 7, 11, 13*, 14, 17*, 19, 21, 22, 23, 29*, 31, 33, 37*, 38, 41*, 43, 46, 47, 53*, 57, 59, 61*, 62, 67, 69, 71, 73*, 77, 83, 86, 89*, 93, 94, 97*, ...Chapter I, section 6, p. 37 of (complete until ''d'' = 100) *: The narrow cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Unit (number Theory)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. ). Real quadratic fields For the real quadratic field K=\mathbf(\sqrt) (with ''d'' square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of ''K'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gross-Zagier Theorem
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one. Gross–Zagier theorem The Gross–Zagier theorem describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point ''s'' = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, showed that Heegner points could be used to construct rational points on the curve for each positive integer ''n'', and the heights of these points were the coefficients of a modular form of weight 3/2. Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 rise to an ''L''-function via analytic continuation. The Riemann zeta function is an example of an ''L''-function, and one important conjecture involving ''L''-functions is the Riemann hypothesis and its generalization. The theory of ''L''-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the ''L''-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between ''L''-functions and the theory of prime numbers. The mathematical field that studies L-func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia. He has held positions at the University of California at Berkeley (Miller Fellow, 1969–1971), Hebrew University (1971–1972), Tel Aviv University (1972–1973), Institute for Advanced Study (1973–1974), in Italy (1974–1976), at MIT (1976–1982), University of Texas at Austin (1983–1985) and Harvard (1982–1985). Since 1985, he has been a professor at Columbia University. He is a member of the editorial board of ''Acta Arithmetica'' and of '' The Ramanujan Journal''. On January 1, 2018 he became the Editor-in-Chief of the Journal of Number Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
