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Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and diophantine geometry ( Hasse principle), and to local zeta functions. Life Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse, also written Haße (12 April 1868 – 1 June 1940, son of Friedrich Ernst Hasse and his wife Anna Von Reinhard) and his wife Margarethe Louise Adolphine Quentin (born 5 July 1872 in Milwaukee, daughter of retail toy merchant Adolph Quentin (b. May 1832, probably Berlin, Kingdom of Prussia) and Margarethe Wehr (b. about 1840, Prussia), then raised in Kassel). After serving in the Imperial German Navy in World War I, he studied at the University of Göttingen, and then at the University of Marburg under Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Mink ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kassel
Kassel (; in Germany, spelled Cassel until 1926) is a city on the Fulda River in North Hesse, northern Hesse, in Central Germany (geography), central Germany. It is the administrative seat of the Regierungsbezirk Kassel (region), Kassel and the district Kassel (district), of the same name, and had 201,048 inhabitants in December 2020. The former capital of the States of Germany, state of Hesse-Kassel, it has many palaces and parks, including the Bergpark Wilhelmshöhe, which is a UNESCO World Heritage Site. Kassel is also known for the ''documenta'' Art exhibition, exhibitions of contemporary art. Kassel has a Public university, public University of Kassel, university with 25,000 students (2018) and a multicultural population (39% of the citizens in 2017 had a migration background). History Kassel was first mentioned in 913 AD, as the place where two deeds were signed by King Conrad of Franconia, Conrad I. The place was called ''Chasella'' or ''Chassalla'' and was a fortifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Schilling
Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether. After Noether was forced to leave Germany by the Nazis, he found a new advisor in Helmut Hasse,. and obtained his Ph.D. from Marburg University in 1934 on the thesis ''Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper''. He then was post doc at Trinity College, Cambridge before moving to Institute for Advanced Study 1935–37 and the Johns Hopkins University 1937–39. He became an instructor with the University of Chicago in 1939, promoted to assistant professor 1943, associate 1945 and full professor in 1958. In 1961 he moved to Purdue University. He died in Highland Park, Illinois. His students were, among others, the game theorist Anatol Rapoport an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milwaukee
Milwaukee is the List of cities in Wisconsin, most populous city in the U.S. state of Wisconsin. Located on the western shore of Lake Michigan, it is the List of United States cities by population, 31st-most populous city in the United States and the fifth-most populous city in the Midwest with a population of 577,222 at the 2020 United States census, 2020 census. It is the county seat of Milwaukee County, Wisconsin, Milwaukee County. The Milwaukee metropolitan area is the Metropolitan statistical area, 40th-most populous metropolitan area in the U.S. with 1.57 million residents. Founded in the early 19th century and incorporated in 1846, Milwaukee grew rapidly due to its location as a port city. History of Milwaukee, Its history was heavily influenced by German immigrants and it continues to be a Germans in Milwaukee, center for German-American culture, specifically known for Beer in Milwaukee, its brewing industry. The city developed as an industrial powerhouse during the 19t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Zeta Function
In mathematics, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^k\right) where is a Singular point of an algebraic variety, non-singular -dimensional projective algebraic variety over the field with elements and is the number of points of defined over the finite field extension of . Making the variable transformation gives : \mathit (V,t) = \exp \left( \sum_^ N_k \frac \right) as the formal power series in the variable t. Equivalently, the local zeta function is sometimes defined as follows: : (1)\ \ \mathit (V,0) = 1 \, : (2)\ \ \frac \log \mathit (V,t) = \sum_^ N_k t^\ . In other words, the local zeta function with coefficients in the finite field is defined as a function whose logarithmic derivative generates the number of solutions of the equation defining in the degree extension Formulation Given a finite field ''F'', there is, up to isomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the ''p''-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers ''and'' in the ''p''-adic numbers for each prime ''p''. Intuition Given a polynomial equation with rational coefficients, if it has a rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems in Diophantine geometry that are of fundamental importance include: * Mordell–Weil theorem * Roth's theorem * Siegel's theorem * Faltings's theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Class Field Theory
In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the ''p''-adic numbers Q''p'' (where ''p'' is any prime number), or the field of formal Laurent series F''q''((''T'')) over a finite field F''q''. Approaches to local class field theory Local class field theory gives a description of the Galois group ''G'' of the maximal abelian extension of a local field ''K'' via the reciprocity map which acts from the multiplicative group ''K''×=''K''\. For a finite abelian extension ''L'' of ''K'' the reciprocity map induces an isomorphism of the quotient group ''K''×/''N''(''L''×) of ''K''× by the norm group ''N''(''L''×) of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing '' ... [...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 Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History 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 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse–Schmidt Derivation
In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by . Definition For a (not necessarily commutative nor associative) ring ''B'' and a ''B''-algebra ''A'', a Hasse–Schmidt derivation is a map of ''B''-algebras :D: A \to A ![t!/math> taking values in the ring of formal power series">">![t<_a>!.html" ;"title=".html" ;"title="![t">![t!">.html" ;"title="![t">![t!/math> taking values in the ring of formal power series with coefficients in ''A''. This definition is found in several places, such as , which also contains the following example: for ''A'' being the ring of infinitely differentiable functions (defined on, say, R''n'') and ''B''=R, the map :f \mapsto \exp\left(t \frac d \right) f(x) = f + t \frac + \frac 2 \frac + \cdots is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly. Equivalent characterizations shows that a Hasse–Schmidt derivation is equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
