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Greenberg's Conjectures
Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021. Invariants conjecture The first conjecture was proposed in 1976 and concerns Iwasawa invariants. This conjecture is related to Vandiver's conjecture, Leopoldt's conjecture, Birch–Tate conjecture, all of which are also unsolved. The conjecture, also referred to as Greenberg's invariants conjecture, firstly appeared in Greenberg's Princeton University thesis of 1971 and originally stated that, assuming that F is a totally real number field and that F_\infty/F is the cyclotomic \mathbb_p-extension, \lambda(F_\infty/F) = \mu(F_\infty/F) = 0, i.e. the power of p dividing the class number of F_n is bounded as n \rightarrow \infty. Note that if Leopoldt's conjecture holds for F and p, the only \mathbb_p-extension of F is the cyclotomic one (since it is totally real). In 1976, Greenberg expanded the conjecture by providing more examples fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Galois Problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of \mathbb having a particular group as Galois group. These groups include all of degree no greater than . There also are groups known not to have generic polynomials, such as the cyclic group of order . More generally, let be a given finite group, and let be a field. Then the question is this: is there a Galois extension field such that the Galois group of the extension is isomorphic to ? One says that is realizable over if such a field exists. Partial results There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any function field in one variable over the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferrero–Washington Theorem
In algebraic number theory, the Ferrero–Washington theorem, proved first by and later by , states that Iwasawa's μ-invariant vanishes for cyclotomic Z''p''-extensions of abelian algebraic 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 f ...s. History introduced the μ-invariant of a Z''p''-extension and observed that it was zero in all cases he calculated. used a computer to check that it vanishes for the cyclotomic Z''p''-extension of the rationals for all primes less than 4000. later conjectured that the μ-invariant vanishes for any Z''p''-extension, but shortly after discovered examples of non-cyclotomic extensions of number fields with non-vanishing μ-invariant showing that his original conjecture was wrong. He suggested, however, that the conjecture might ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Larry Washington
Lawrence Clinton Washington (born 1951, Vermont) is an American mathematician at the University of Maryland who specializes in number theory. Biography Washington studied at Johns Hopkins University, where in 1971 he received his B.A. and master's degree. In 1974 he earned his PhD at Princeton University under Kenkichi Iwasawa with thesis ''Class numbers and Z_p extensions''. He then became an assistant professor at Stanford University and from 1977 at the University of Maryland, where he became in 1981 an associate professor and in 1986 a professor. He held visiting positions at several institutions, including IHES (1980/81), Max-Planck-Institut für Mathematik (1984), the Institute for Advanced Study (1996), and MSRI (1986/87), as well as at the University of Perugia, Nankai University and the State University of Campinas. In 1979–1981 he was a Sloan Fellow. Recognition He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Greenberg
Ralph Greenberg (born 1944) is an American mathematician who has made contributions to number theory, in particular Iwasawa theory. He was born in Chester, Pennsylvania and studied at the University of Pennsylvania, earning a B.A. in 1966, after which he attended Princeton University, earning his doctorate in 1971 under the supervision of Kenkichi Iwasawa. Greenberg's results include a proof (joint with Glenn Stevens) of the Mazur–Tate–Teitelbaum conjecture as well as a formula for the derivative of a ''p''-adic Dirichlet ''L''-function at s=0 (joint with Bruce Ferrero). Greenberg is also well known for his many conjectures. In his PhD thesis, he conjectured that the Iwasawa μ- and λ-invariants of the cyclotomic \Z_p-extension of a totally real field are zero, a conjecture that remains open as of September 2012. In the 1980s, he introduced the notion of a Selmer group for a ''p''-adic Galois representation and generalized the "main conjectures" of Iwasawa and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Number 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thesis
A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: Documentation�Presentation of theses and similar documents International Organization for Standardization, Geneva, 1986. In some contexts, the word "thesis" or a cognate is used for part of a bachelor's or master's course, while "dissertation" is normally applied to a doctorate. This is the typical arrangement in American English. In other contexts, such as within most institutions of the United Kingdom and Republic of Ireland, the reverse is true. The term graduate thesis is sometimes used to refer to both master's theses and doctoral dissertations. The required complexity or quality of research of a thesis or dissertation can vary by country, university, or program, and the required minimum study period may thus vary significantly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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