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Reflection Theorem
In algebraic number theory, a reflection theorem or Spiegelungssatz (German for ''reflection theorem'' – see ''Spiegel'' and '' Satz'') is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field \mathbb \left( \zeta_p \right), with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield \mathbb \left( \zeta_p \right)^ is. Another example is due to Scholz. A simplified version of his theorem states that if 3 divides the class number of a real quadratic field \mathbb \left( \sqrt \right), then 3 also divides the class number of the imaginary quadratic field \mathbb \left( \sqrt \right). Leopoldt's Spiegelungssatz Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks ... [...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 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 Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinrich-Wolfgang Leopoldt
Heinrich-Wolfgang Leopoldt (22 August 1927 – 28 July 2011) was a German mathematician who worked on algebraic number theory. Leopoldt earned his Ph.D. in 1954 at the University of Hamburg under Helmut Hasse with the thesis ''Über Einheitengruppe und Klassenzahl reeller algebraischer Zahlkörper'' (''On group of unity and class number of real algebraic number fields''). As a postdoc, he was from 1956 to 1958 at the Institute for Advanced Study. In 1959, he obtained his habilitation degree at the University of Erlangen and was then at the University of Tübingen. From 1964, he was ordentlicher Professor at the University of Karlsruhe, where he was also Director of the Mathematics Institute. Leopoldt and Tomio Kubota introduced and investigated p-adic L-functions (now named after them). These functions are a component of Iwasawa theory and are a p-adic version of the Dirichlet L-functions. With Hans Zassenhaus he also worked on computer algebra and its applications in number t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives. Formulation Iwasawa worked with so-called \Z_p-extensions - infinite extensions of a number field F with Galois group \Gamma isomorphic to the additive group of p-adic integers for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\Z. Iwasawa studied classical Galois modules over F_n by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenkichi Iwasawa
Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. Biography Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture. He attended elementary school there, but later moved to Tokyo to attend Musashi High School. From 1937 to 1940 Iwasawa studied as an undergraduate at Tokyo Imperial University, after which he entered graduate school at University of Tokyo and became an assistant in the Department of Mathematics. In 1945 he was awarded a Doctor of Science degree. However, this same year Iwasawa became sick with pleurisy, and was unable to return to his position at the university until April 1947. From 1949 to 1955 he worked as assistant professor at Tokyo University. In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Gras
Georges may refer to: Places *Georges River, New South Wales, Australia *Georges Quay (Dublin) * Georges Township, Fayette County, Pennsylvania Other uses *Georges (name) * ''Georges'' (novel), a novel by Alexandre Dumas * "Georges" (song), a 1977 song originally recorded by Pat Simon and covered by Sylvie Vartan *Georges (store), a department store in Melbourne, Australia from 1880 to 1995 * Georges (''Green Card'' character) People with the surname *Eugenia Georges, American anthropologist *Karl Ernst Georges (1806–1895), German classical philologist and lexicographer, known for his edition of Latin-German dictionaries. See also *École secondaire Georges-P.-Vanier, a high school in Hamilton, Ontario, Canada *École secondaire Georges-Vanier in Laval, Quebec, Canada * French cruiser ''Georges Leygues'', commissioned in 1937 * French frigate ''Georges Leygues'' (D640), commissioned in 1979 *George (other) *Georges Creek (other) *Georges Creek Coal and Iron Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ''v'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolved prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Language
German ( ) is a West Germanic languages, West Germanic language mainly spoken in Central Europe. It is the most widely spoken and Official language, official or co-official language in Germany, Austria, Switzerland, Liechtenstein, and the Italy, Italian province of South Tyrol. It is also a co-official language of Luxembourg and German-speaking Community of Belgium, Belgium, as well as a national language in Namibia. Outside Germany, it is also spoken by German communities in France (Bas-Rhin), Czech Republic (North Bohemia), Poland (Upper Silesia), Slovakia (Bratislava Region), and Hungary (Sopron). German is most similar to other languages within the West Germanic language branch, including Afrikaans, Dutch language, Dutch, English language, English, the Frisian languages, Low German, Luxembourgish, Scots language, Scots, and Yiddish. It also contains close similarities in vocabulary to some languages in the North Germanic languages, North Germanic group, such as Danish lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolved prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
