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Artin–Zorn Theorem
In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin. The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the proper ... is the classical projective plane over a finite field.. References Theorems in ring theory {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrians, Austrian and Armenian people, Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as "opera singer" though others list it as "art dealer." It see ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Zorn
Max August Zorn (; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, and ordered sets amongst others. Zorn's lemma was first postulated by Kazimierz Kuratowski in 1922, and then independently by Zorn in 1935. Life and career Zorn was born in Krefeld, Germany. He attended the University of Hamburg. He received his PhD in April 1930 for a thesis on alternative algebras. He published his findings in '' Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg''. Zorn showed that split-octonions could be represented by a mixed-style of matrices called Zorn's vector-matrix algebra. Max Zorn was appointed to an assistant position at the University of Halle. However, he did not have the opportunity to work there for long as he was forced to leave Germany ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Division Ring
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to : ,x,y= 0 : ,x,x= 0 Both of these identities together imply that: : ,y,x ,x,x ,y,x :- ,x+y,x+y= := ,x+y,-y= := ,x,-y- ,y,y= 0 for all x and y. This is equivalent to the '' flexible identity'' :(xy)x = x(yx). The associator of an alternative algebra is therefore alternating. ... [...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] |
Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg
(English: ''Reports from the Mathematical Seminar of the University of Hamburg'') is a peer-reviewed mathematics journal published by Springer Science+Business Media. It publishes articles on pure mathematics and is scientifically coordinated by the ''Mathematisches Seminar'', an informal cooperation of mathematicians at the Universität Hamburg; its managing editors are professors and Tobias Dyckerhoff. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. History The ''Abhandlungen'' were set up as a new journal by Wilhelm Blaschke in 1922 at the newly created Department of Mathematics (called ''Mathematisches Seminar'') at the newly founded Hamburgische Universität. Blaschke invited Hermann Weyl and David Hilbert to the ''Mathematisches Seminar'' (in 1920 and 1921, respectively) to deliver a talk series on their views concerning the Foundations of Mathematics. These talks formed part of the early history of the Grundlagenkrise der Mathematik, and Hilbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedderburn Theorem
In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field. History The original proof was given by Joseph Wedderburn in 1905,Lam (2001), p. 204/ref> who went on to prove the theorem in two other ways. Another proof was given by Leonard Eugene Dickson shortly after Wedderburn's original proof, and Dickson acknowledged Wedderburn's priority. However, as noted in , Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs appeared only after he had read Dickson's correct proof. On this basis, Parshall argues that Dickson should be credited with the first correct proof. A simplified version of the proof was later given by Ernst Witt. Witt's proof is sketched ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moufang Plane
In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line. A translation plane is Moufang if every line of the plane is a translation line. Characterizations A Moufang plane can also be described as a projective plane in which the '' little Desargues theorem'' holds. This theorem states that a restricted form of Desargues' theorem holds for every line in the plane. For example, every Desarguesian plane is a Moufang plane. In algebraic terms, a projective plane over any alternative division ring is a Moufang plane, and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and of Moufang planes. As a consequence of the algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
