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Level (algebra)
In field theory (mathematics), field theory, a branch of mathematics, the Stufe (/Help:IPA for German, ʃtuːfə/; German: level) ''s''(''F'') of a field (mathematics), field ''F'' is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, ''s''(''F'') = \infty. In this case, ''F'' is a formally real field. Albrecht Pfister (mathematician), Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs. Powers of 2 If s(F)\ne\infty then s(F)=2^k for some natural number k.Rajwade (1993) p.13Lam (2005) p.379 ''Proof:'' Let k \in \mathbb N be chosen such that 2^k \leq s(F) < 2^. Let . Then there are elements such that : Both and are sums of squares, and , since otherwise |
Field Theory (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponent Of A Group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent that divides its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich (see '' Golod–Shafarevich theorem''), and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C (with their standard topologies) are Archimedean local fields. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Quarterly
The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; by Clark Kimberling the present editor is Professor Curtis Cooper of the Mathematics Department of the University of Central Missouri. The ''Fibonacci Quarterly'' has an editorial board of nineteen members and is overseen by the nine-me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratically Closed Field
In mathematics, a quadratically closed field is a field of characteristic not equal to 2 in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples * The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed. * The field of real numbers is not quadratically closed as it does not contain a square root of −1. * The union of the finite fields \mathbb F_ for ''n'' ≥ 0 is quadratically closed but not algebraically closed. * The field of constructible numbers is quadratically closed but not algebraically closed.Lam (2005) p. 220 Properties * A field is quadratically closed if and only if it has universal invariant equal to 1. * Every quadratically closed field is a Pythagorean field but not conversely (for example, R is Pythagorean); however, every non- formally real Pythagorean field is quadratically closed. * A field is quadratically closed if and only if its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic not equal to 2. All vector spaces will be assumed to be finite- dimensional. Two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p. 14 Each class is represented by the core form of a Witt decomposition.Lorenz (2008) p. 30 The Witt group of ''k'' is the abelian group ''W''(''k'') of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms.Milnor & Husemoller (1973) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
