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Field Arithmetic
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups. Fields with finite absolute Galois groups Let ''K'' be a field and let ''G'' = Gal(''K'') be its absolute Galois group. If ''K'' is algebraically closed, then ''G'' = 1. If ''K'' = R is the real numbers, then :G=\operatorname(\mathbf/\mathbf)=\mathbf/2 \mathbf. Here C is the field of complex numbers and Z is the ring of integer numbers. A theorem of Artin and Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups. Artin–Schreier theorem. Let ''K'' be a field whose absolute Galois group ''G'' is finite. Then either ''K'' is separably closed and ''G'' is trivial or ''K'' is real closed and ''G'' = Z/2Z ... [...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] |
David Harbater
David Harbater (born December 19, 1952) is an American mathematician at the University of Pennsylvania, well known for his work in Galois theory, algebraic geometry and arithmetic geometry. Early life and education Harbater was born in New York City and attended Stuyvesant High School, where he was on the math team. After graduating in 1970, he entered Harvard University. After graduating summa cum laude in 1974, Harbater earned a master's degree from Brandeis University and then a Ph.D. in 1978 from MIT, where he wrote a dissertation (Deformation Theory and the Fundamental Group in Algebraic Geometry) under the direction of Michael Artin. Research He solved the inverse Galois problem over \mathbb_p(t), and made many other significant contributions to the field of Galois theory. Harbater's recent work on patching over fields, together with Julia Hartmann and Daniel Krashen, has had applications in such varied fields as quadratic forms, central simple algebras and local-global p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Fried (mathematician)
Michael David Fried is an American mathematician working in the geometry and arithmetic of families of nonsingular projective curve covers. Career Fried received his undergraduate degree from Michigan State University in electrical engineering and then worked for three years as an aerospace electrical engineer. He then received his PhD from University of Michigan in Mathematics in 1967 under Donald John Lewis. He spent two years as a postdoctoral researcher at the Institute for Advanced Study (1967–1969). He was a professor at Stony Brook University (8 years), University of California at Irvine (26 years), University of Florida (3 years) and Hebrew University (2 years). He has held visiting appointments at MIT, MSRI, University of Michigan, University of Florida, Hebrew University and Tel Aviv University. He has been an editor on several mathematics journals including the Research Announcements of the Bulletin of the American Mathematical Society, and the Journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Roquette
Peter Roquette (born 8 October 1927) is a German mathematician working in algebraic geometry, algebra, and number theory. He was born in Königsberg. Roquette studied in Erlangen, Berlin, and Hamburg. In 1951 he defended a dissertation at the University of Hamburg under Helmut Hasse, providing a new proof of the Riemann hypothesis for algebraic function fields over a finite field (the first proof was given by André Weil in 1940). In 1951/1952 he was an assistant at the Mathematical Research Institute at Oberwolfach and from 1952 to 1954 at the University of Munich. From 1954 to 1956 he worked at the Institute for Advanced Study in Princeton. In 1954 he was Privatdozent at Munich, and from 1956 to 1959 he worked in the same position at Hamburg. In 1959 he became an associate professor at the University of Saarbrucken and in the same year at the University of Tübingen. From 1967 he was professor at the Ruprecht-Karls-University of Heidelberg, where he retired in 1996. Roquette wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding Problem
In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given. Definition Given a field ''K'' and a finite group ''H'', one may pose the following question (the so called inverse Galois problem). Is there a Galois extension ''F/K'' with Galois group isomorphic to ''H''. The embedding problem is a generalization of this problem: Let ''L/K'' be a Galois extension with Galois group ''G'' and let ''f'' : ''H'' → ''G'' be an epimorphism. Is there a Galois extension ''F/K'' with Galois group ''H'' and an embedding ''α'' : ''L'' → ''F'' fixing ''K'' under which the restriction map from the Galois group of ''F/K'' to the Galois group of ''L/K'' coincides with ''f''? Analogously, an embedding problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbertian Field
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within ''K'' a polynomial that does not always factorise. One is also allowed to take finite unions. Formulation More precisely, let ''V'' be an algebraic variety over ''K'' (assumptions here are: ''V'' is an irreducible set, a quasi-projective variety, and ''K'' has characteristic zero). A type I thin set is a subset of ''V''(''K'') that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than ''d'', the dimension of ''V''. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the ''K''-points of some other ''d''-dimensional al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field ''k'', and an algebraically closed extension ''K'' of ''k'', an affine variety ''X'' over ''k'' is the set of common zeros in K^n of a collection of polynomials with coefficients in ''k'': :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the ''points'' of ''X''. A ''k''-rational point (or ''k''-point) of ''X'' is a point of ''X'' that belongs to ''k''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Irreducible
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers and the reals, it is reducible over the complex numbers as x^2+y^2 = (x+iy)(x-iy), and thus not absolutely irreducible. More generally, a polynomial defined over a field ''K'' is absolutely irreducible if it is irreducible over every algebraic extension of ''K'', and an affine algebraic set defined by equations with coefficients in a field ''K'' is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of ''K''. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety,. which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo Algebraically Closed Field
In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.Fried & Jarden (2008) p.218 Formulation A field ''K'' is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds: *Each absolutely irreducible variety V defined over K has a K-rational point. *For each absolutely irreducible polynomial f\in K _1,T_2,\cdots ,T_r,X/math> with \frac\not =0 and for each nonzero g\in K _1,T_2,\cdots ,T_r/math> there exists (\textbf,b)\in K^ such that f(\textbf,b)=0 and g(\textbf)\not =0. *Each absolutely irreducible polynomial f\in K ,X/math> has infinitely many K-rational points. *If R is a finitely generated integral domain over K with quotient field which is regular over K, then there exist a homomorphism h:R\to K such that h(a) = a for each a \in K. Examples * Algebraically closed fields and separably closed fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koji Uchida , a fungus used in East Asian fermentation
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Koji, Kōji, Kohji or Kouji may refer to: * Kōji (given name), a masculine Japanese given name * Kōji (Heian period) (康治), Japanese era, 1142–1144 * Kōji (Muromachi period) (弘治), Japanese era, 1555–1558 * Koji orange, a Japanese citrus cultivar * Andrew Koji Shiraki (born 1987), singer/songwriter known as ''Koji'' *Koji, the software that builds RPM packages for the Fedora project *''Koji'', the common name of the fungus ''Aspergillus oryzae'' *Koji, an interactive content creation tool from GoMeta See also *Kojii, music project by Kojii Helnwein *'' Coji-Coji'' (コジコジ), an anime series sometimes romanized ''Koji Koji'' *Kōji mold Aspergillus oryzae ''Aspergillus oryzae'', also known as , is a filamentous fungus (a mold) used in East Asia to saccharify rice, sweet potato, and barley in the making of alcoholic beverages such as ''sake'' and '' shōchū'', and also to ferment soybeans for m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Neukirch
Jürgen Neukirch (24 July 1937 – 5 February 1997) was a German mathematician known for his work on algebraic number theory. Education and career Neukirch received his diploma in mathematics in 1964 from the University of Bonn. For his Ph.D. thesis, written under the direction of Wolfgang Krull, he was awarded in 1965 the Felix-Hausdorff-Gedächtnis-Preis. He completed his habilitation one year later. From 1967 to 1969 he was guest professor at Queen's University in Kingston, Ontario and at the Massachusetts Institute of Technology in Cambridge, Massachusetts, after which he was a professor in Bonn. In 1971 he became a professor at the University of Regensburg. Contributions He is known for his work on the embedding problem in algebraic number theory, the Báyer–Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch–Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
