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Jacobson–Bourbaki Theorem
In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend Galois theory to field extensions that need not be separable. It was introduced by for commutative field (mathematics), fields and extended to non-commutative fields by , and who credited the result to unpublished work by Nicolas Bourbaki. The extension of Galois theory to normal extensions is called the Jacobson–Bourbaki correspondence, which replaces the correspondence between some Field extension, subfields of a field and some subgroups of a Galois group by a correspondence between some sub division rings of a division ring and some subalgebras of an associative algebra. The Jacobson–Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a purely inseparable extension of exponent at most 1. Statement Suppose that ''L'' is a division ring. The Jacobson–Bourbaki theorem states that there is a natural 1:1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. 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 fields of rational functions, algebraic function fields, algebraic number fields, and ''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 elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in mathematical analysis, analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Carta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. Definition Let ''L/K'' be an algebraic extension (i.e. ''L'' is an algebraic extension of ''K''), such that L\subseteq \overline (i.e. ''L'' is contained in an algebraic closure of ''K''). Then the following conditions, any of which can be regarded as a definition of ''normal extension'', are equivalent: * Every embedding of ''L'' in \overline induces an automorphism of ''L''. * ''L'' is the splitting field of a family of polynomials in K\left \right/math>. * Every irreducible polynomial of K\left \right/math> which has a root in ''L'' splits into linear factors in ''L''. Other properties Let ''L'' be an extension of a field ''K''. Then: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...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] |
Division Ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have . A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). All division rings are simple. That is, they have no two-sided ideal besi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of ''all'' algebraic structures. "Subalgebra" can refer to either case. Subalgebras for algebras over a ring or field A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purely Inseparable Extension
In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x''''q'' = ''a'', with ''q'' a power of ''p'' and ''a'' in ''k''. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. Purely inseparable extensions An algebraic extension E\supseteq F is a ''purely inseparable extension'' if and only if for every \alpha\in E\setminus F, the minimal polynomial of \alpha over ''F'' is ''not'' a separable polynomial.Isaacs, p. 298 If ''F'' is any field, the trivial extension F\supseteq F is purely inseparable; for the field ''F'' to possess a ''non-trivial'' purely inseparable extension, it must be imperfect as outlined in the above section. Several equivalent and more concrete definitions for the notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multilingual journals Multidisciplinary scientific journals Société Mathématique de France academic journals {{mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
