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List Of Things Named After Évariste Galois
The following is a list of topics named after Évariste Galois (1811–1832), a French mathematician. Mathematics *Galois closure *Galois cohomology *Galois connection **Covering_space#Galois_correspondence, Galois correspondence *Galois/Counter Mode *Galois covering *Galois deformation *Galois descent *Galois extension *Galois field *Galois geometry *Galois group **Absolute Galois group *Linear feedback shift register#Galois LFSRs, Galois LFSRs *Galois module *Galois representation *Galois ring *Galois theory **Differential Galois theory **Topological Galois theory *Inverse Galois problem Other *Galois (crater) {{DEFAULTSORT:List of things named after Evariste Galois Lists of things named after mathematicians, Galois ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. Galois was a staunch French Republicans under the Restoration, republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, Galois fought in a duel and died of the wounds he suffered. Life Early life Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a First French Republi ... [...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, then \op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Galois Problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of \mathbb having a particular group as Galois group. These groups include all of degree no greater than . There also are groups known not to have generic polynomials, such as the cyclic group of order . More generally, let be a given finite group, and a field. If there is a Galois extension field whose Galois group is isomorphic to , one says that is realizable over . Partial results Many cases are known. It is known that every finite group is realizable over any function field in one variable over the complex numbers \mathbb, and more generally over function fields in one variable over any algebraically closed field of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Galois Theory
In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by Vladimir Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory. It connects many ideas from algebra to ideas in topology. As described in Askold Khovanskii's book: "According to this theory, the way the Riemann surface of an analytic function covers the plane of complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...s can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way." References * * * Galois theory Topology {{Topology-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Galois Theory
In mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of field (mathematics), algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation (abstract algebra), derivation, ''D''. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. Motivation and basic concepts In mathematics, some types of elementary functions cannot express the indefinite integrals of other elementary functions. A well-known example is e^, whose indefinite integral is the error function \operatornamex, familiar in statistics. Other examples include the sinc function \tfrac and x^x. It's import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), 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 root of a function, 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 by definition ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, nth root, 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Ring
In mathematics, Galois rings are a type of finite commutative rings which generalize both the finite fields and the rings of integers modulo a prime power. A Galois ring is constructed from the ring \mathbb/p^n\mathbb similar to how a finite field \mathbb_ is constructed from \mathbb_p. It is a Galois extension of \mathbb/p^n\mathbb, when the concept of a Galois extension is generalized beyond the context of fields. Galois rings were studied by Krull (1924), and independently by Janusz (1966) and by Raghavendran (1969), who both introduced the name ''Galois ring''. They are named after Évariste Galois, similar to ''Galois fields'', which is another name for finite fields. Galois rings have found applications in coding theory, where certain codes are best understood as linear codes over \Z / 4\Z using Galois rings GR(4, ''r''). Definition A Galois ring is a commutative ring of characteristic ''p''''n'' which has ''p''''nr'' elements, where ''p'' is prime and ''n'' and ''r'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Representation
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ''v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Feedback Shift Register
In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. The most commonly used linear function of single bits is exclusive-or (XOR). Thus, an LFSR is most often a shift register whose input bit is driven by the XOR of some bits of the overall shift register value. The initial value of the LFSR is called the seed, and because the operation of the register is deterministic, the stream of values produced by the register is completely determined by its current (or previous) state. Likewise, because the register has a finite number of possible states, it must eventually enter a repeating cycle. However, an LFSR with a well-chosen feedback function can produce a sequence of bits that appears random and has a very long cycle. Applications of LFSRs include generating pseudo-random numbers, pseudo-noise sequences, fast digital counters, and whitening sequences. Both hardware and software implementations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R, and its degree over R is ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group of profinite integers :: \hat = \varprojlim \mathbf/n\mathbf. :(For the notation, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Geometry
Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Galois geometry may be defined as a projective space over a finite field. Objects of study include affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries. Vector spaces defined over finite fields play a significant role, especially in construction methods. Projective spaces over finite fields Notation Although the generic notation of projective geometry is sometimes used, it is more common to denote projective spaces over finite fields by , where is the "geometric" dimension (see below), and is the order of the finite field (or Galois ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
