Stufe (algebra)

	Stufe (algebra) In field theory, a branch of mathematics, the Stufe (/ ʃtuːfə/; German: level) ''s''(''F'') of a 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 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 $n = 2^k$ . Then there are $s = s(F)$ elements $e_1, \ldots, e_s \in F\setminus\$ such that : $0 = \underbrace_ + \underbrace_\;.$ Both $a$ and $b$ are sums of $n$ squares, and $a \ne 0$ , since otherwise $s(F)< 2^k$ , contrary to the assumption on $k$ . According to the theory of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Field Theory (mathematics) Field theory may refer to: Science * Field (mathematics), the theory of the algebraic concept of field * Field theory (physics), a physical theory which employs fields in the physical sense, consisting of three types: Classical field theory, the theory and dynamics of classical fields Quantum field theory, the theory of quantum mechanical fields Statistical field theory, the theory of critical phase transitions Grand unified theory Social science * Field theory (psychology) Field theory is a psychological theory (more precisely: Topological and vector psychology) which examines patterns of interaction between the individual and the total field, or environment. The concept first made its appearance in psychology with r ..., a psychological theory which examines patterns of interaction between the individual and his or her environment * Field theory (sociology), a sociological theory concerning the relationship between social actors and local social orders {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Pythagoras Number In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number ''p''(''K'') of a field ''K'' is the smallest positive integer ''p'' such that every sum of squares in ''K'' is a sum of ''p'' squares. A ''Pythagorean field'' is a field with Pythagoras number 1: that is, every sum of squares is already a square. Examples * Every non-negative real number is a square, so ''p''(R) = 1. * For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares,Lam (2005) p. 36 so ''p'' = 2. * By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so ''p''(Q) = 4. Properties * Every positive integer occurs as the Pythagoras number of some formally real field.Lam (2005) p. 398 * The Pythagoras number is related to the Stufe by ''p''(''F'') ≤ ''s''( ... [...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]
	Ergebnisse Der Mathematik Und Ihrer Grenzgebiete ''Ergebnisse der Mathematik und ihrer Grenzgebiete''/''A Series of Modern Surveys in Mathematics'' is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related areas". Most of the books were published in German or English, but there were a few in French and Italian. There have been several sequences, or ''Folge'': the original series, neue Folge, and 3.Folge. Some of the most significant mathematical monographs of 20th century appeared in this series. Original series The series started in 1932 with publication of ''Knotentheorie'' by Kurt Reidemeister as "Band 1" (English: volume 1). There seems to have been double numeration in this sequence. Neue Folge This sequence started in 1950 with the publication of ''Transfinite Zahlen'' by Heinz Bachmann. The volumes are consecutively numbered, designated as either "Band" or "Heft". A total of 100 volumes was published, often in multiple editions, but pre ... [...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 (199 ... [...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 topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. 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 lea ... [...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;Biography of Verner Emil Hoggatt Jr. 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 an ... [...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 that contains a finite number of elements. As with any field, a finite field is a 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 given by the integers mod when 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 and every positive integer there are fields of order p^k, all of which are 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 which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...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, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual 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. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Quadratically Closed Field In mathematics, a quadratically closed field is a field 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 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 Witt–Grothendieck ring is isomorphic t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Witt Group In mathematics, a Witt group of a field (mathematics), field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear form, symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic (algebra), characteristic not equal to two. All vector spaces will be assumed to be finite-dimension (vector space), dimensional. We say that 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 (quadratic forms), 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 correspo ... [...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 dividing 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 finite exponent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]