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Principal Ideal Theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called Principalization (algebra), ''principalization'', or sometimes ''capitulation''. Formal statement For any algebraic number field ''K'' and any ideal (ring theory), ideal ''I'' of the ring of integers of ''K'', if ''L'' is the Hilbert class field of ''K'', then :IO_L\ is a principal ideal α''O''''L'', for ''O''''L'' the ring of integers of ''L'' and some element α in it. History The principal ideal theorem was conjectured by , and was the last remaining aspect of his program on class fields to be completed, in 1929. reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the tran ... [...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] |
Transfer (group Theory)
In the mathematical field of group theory, the transfer defines, given a group ''G'' and a subgroup of finite index ''H'', a group homomorphism from ''G'' to the abelianization of ''H''. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups. The transfer was defined by and rediscovered by . Construction The construction of the map proceeds as follows:Following Scott 3.5 Let 'G'':''H''= ''n'' and select coset representatives, say :x_1, \dots, x_n,\, for ''H'' in ''G'', so ''G'' can be written as a disjoint union :G = \bigcup\ x_i H. Given ''y'' in ''G'', each ''yxi'' is in some coset ''xjH'' and so :yx_i = x_jh_i for some index ''j'' and some element ''h''''i'' of ''H''. The value of the transfer for ''y'' is defined to be the image of the product :\textstyle \prod_^n h_i in ''H''/''H''′, where ''H''′ is the commutator subgroup of ''H''. The order of the factors is irrelevant since ''H''/''H''†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideals (ring Theory)
Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered in abstract algebra * Ideal, special subsets of a semigroup * Ideal (order theory), special kind of lower sets of an order * Ideal (set theory), a collection of sets regarded as "small" or "negligible" * Ideal (Lie algebra), a particular subset in a Lie algebra * Ideal point, a boundary point in hyperbolic geometry * Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points Science * Ideal chain, in science, the simplest model describing a polymer * Ideal gas law, in physics, governing the pressure of an ideal gas * Ideal transformer, an electrical transformer having zero resistance and perfect magnetic threading * Ideal final result, in TRIZ methodology, the best possible solution * Thought experiment, sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Fields
''Corps Locaux'' by Jean-Pierre Serre, originally published in 1962 and translated into English as ''Local Fields'' by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, group cohomology, and 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 re .... The book's end goal is to present local class field theory from the cohomological point of view. This theory concerns extensions of "local" (i.e., complete for a discrete valuation) fields with finite residue field. Contents #''Part I, Local Fields (Basic Facts)'': Discrete valuation rings, Dedekind domains, and Completion. #''Part II, Ramification'': Discriminant & Different, Ramification Groups, The Norm, and Artin Representation. #''Part I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philipp Furtwängler
Friederich Pius Philipp Furtwängler (April 21, 1869 – May 19, 1940) was a German number theorist. Biography Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (''Zur Theorie der in Linearfaktoren zerlegbaren ganzzahligen ternären kubischen Formen''), under Felix Klein. Most of his academic life, from 1912 to 1938, was spent at the University of Vienna, where he taught for example Kurt Gödel, who later said that Furtwängler's lectures on number theory were the best mathematical lectures that he ever heard; Gödel had originally intended to become a physicist but turned to mathematics partly as a result of Furtwängler's lectures. Furtwängler was paralysed and, without notes, lectured from a wheelchair while his assistant wrote equations on the blackboard. Some of Furtwängler's doctoral students were Wolfgang Gröbner, Nikolaus Hofreiter, Henry Mann, Otto Schreier, and Olga Taussky-Todd. Through these and others, he has over 30 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept. Definitions * a ''left principal ideal'' of R is a subset of R given by Ra = \ for some element a, * a ''right principal ideal'' of R is a subset of R given by aR = \ for some element a, * a ''two-sided principal ideal'' of R is a subset of R given by RaR = \ for some element a, namely, the set of all finite sums of elements of the form ras. While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition. If R is a commuta ... [...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 ''CF' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often denoted by O_K or \mathcal O_K. Since any integer belongs to K and is an integral element of K, the ring \mathbb is always a subring of O_K. The ring of integers \mathbb is the simplest possible ring of integers. Namely, \mathbb=O_ where \mathbb is the field of rational numbers. And indeed, in algebraic number theory the elements of \mathbb are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers \mathbb /math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field \mathbb(i) of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
