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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 cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Fields
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 field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich Martin Weber
Heinrich Martin Weber (5 March 1842, Heidelberg, German Confederation, 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 Mathematical analysis, analysis. He is best known for his text ''Lehrbuch der Algebra'' published in 1895 and much of it is his original research in algebra and number theory. His work ''Theorie der algebraischen Functionen einer Veränderlichen'' (with Richard Dedekind, Dedekind) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann–Roch theorem. Weber's research papers were numerous, most of them appearing in ''Crelle's Journal'' or ''Mathematische Annalen''. He was the editor of Bernhard Riemann, Riemann's collected works. Weber was born in Heidelberg, Grand Duchy of Baden, Baden, and entered the University of Heidelberg in 1860. In 1866 he became a privatdozent, and in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer–Siegel Theorem
In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the results known on the class numbers of imaginary quadratic fields, to a more general sequence of number fields :K_1, K_2, \ldots.\ In all cases other than the rational field Q and imaginary quadratic fields, the regulator ''R''''i'' of ''K''''i'' must be taken into account, because ''K''i then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if ''D''''i'' is the discriminant of ''K''''i'', then : \frac \to 0\texti \to\infty. Assuming that, and the algebraic hypothesis that ''K''''i'' is a Galois extension of Q, the conclusion is that : \frac \to 1\texti \to\infty where ''h''''i'' is the class number of ''K''''i''. If one assumes that all the degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Number Formula
In number theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function. General statement of the class number formula We start with the following data: * is a number field. * , where denotes the number of real embeddings of , and is the number of complex embeddings of . * is the Dedekind zeta function of . * is the class number, the number of elements in the ideal class group of . * is the regulator of . * is the number of roots of unity contained in . * is the discriminant of the extension . Then: :Theorem (Class Number Formula). converges absolutely for and extends to a meromorphic function defined for all complex with only one simple pole at , with residue :: \lim_ (s-1) \zeta_K(s) = \frac This is the most general "class number formula". In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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). Gauss discusses imaginary quadratic fields in Article 303, stating the first two conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositio Mathematica
''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the ''Journal Citation Reports'', the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Fabrizio Andreatta, David Holmes, Bruno Klingler, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from ''Mathematische Annalen'' in 1928. An announcement of the new journal was made in a 1934 issue of the ''American Mathematical Monthly''. In 1940, the publication of the journal was suspended due to the German occupation of the Netherlands Despite Dutch neutrality, Nazi Germany German invasion of the Netherlands, invaded the Netherlands ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish- American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in 1975 at Bell Telephone Laboratories, where he stayed for 26 years before joining the University of Minnesota in 2001. Work in mathematics Odlyzko received his B.S. and M.S. in mathematics from the California Institute of Technology and his Ph.D. from the Massachusetts Institute of Technology in 1975. In the field of mathematics he has published extensively on analytic number theory, computational number theory, cryptography, algorithms and computational complexity, combinatorics, probability, and error-correcting codes. In the early 1970s, he was a co-author (with D. Kahaner and Gian-Carlo Rota) of one of the founding papers of the modern umbral calculus. In 1985 he and Herman te Riele disproved the Mertens conjecture. In mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Stark
Harold Mead Stark (born August 6, 1939) is an Americans, American mathematician, specializing in number theory. He is best known for his solution of the Carl Friedrich Gauss, Gauss class number 1 problem, in effect Stark–Heegner theorem, correcting and completing the earlier work of Kurt Heegner, and for Stark's conjecture. More recently, he collaborated with Audrey Terras to study Ihara zeta function, zeta functions in graph theory. He is currently on the faculty of the University of California, San Diego. Stark received his bachelor's degree from the California Institute of Technology in 1961 and his PhD from the University of California, Berkeley in 1964. He was on the faculty at the University of Michigan from 1964 to 1968, at the Massachusetts Institute of Technology from 1968 to 1980, and at the University of California, San Diego from 1980 to the present. Stark was elected to the American Academy of Arts and Sciences in 1983 and to the United States National Academy of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the tensor product algebra of ''F'' with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial ''P'' irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Imaginary Number Field
In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...s, and, more generally, CM fields. Any number field that is Galois over the rationals must be either totally real or totally imaginary. References *Section 13.1 of Algebraic number theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John H
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died ), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (died ), one of the twelve apostles of Jesus Christ * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
