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# Topic: Galois theory

 Galois Theory for Beginners Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular n-gons are also presented. Using the class of algebraic objects that we previously mentioned, it became possible at the beginning of the twentieth century to reformulate what has come to be called Galois theory, and indeed in such a way that the problem itself can be posed in terms of such objects. www.galois-theorie.de /galois-theory.htm   (1004 words)

 Galois theory - Wikipedia, the free encyclopedia Further abstraction of Galois theory is achieved by the theory of Galois connections. Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. en.wikipedia.org /wiki/Galois_theory   (1607 words)

 Galois theory   (Site not responding. Last check: 2007-10-22) Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. Symmetries are usually expressed in terms of symmetry groups, and in fact the abstract notion of group was invented by Evariste Galois for the very purpose of describing symmetries of roots. In the modern approach, the formalism is changed somewhat, in order to achieve a precise and more general definition: one starts with a field extension L/K and defines its Galois group as the group of all field automorphisms of L which keep all elements of K fixed. www.worldwidewebfind.com /encyclopedia/en/wikipedia/g/ga/galois_theory.html   (667 words)

 [No title]   (Site not responding. Last check: 2007-10-22) Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. www.worldhistory.com /wiki/G/Galois-theory.htm   (632 words)

 Galois theory   (Site not responding. Last check: 2007-10-22) In mathematics Galois theory is that branch of abstract algebra which studies the symmetries of the of polynomials. In other words the Galois theory the study of solutions to polynomials and the different solutions are related to each Symmetries are usually expressed in terms of symmetry groups and in fact the very notion a group was invented by Evariste Galois to describe symmetries of roots. Galois Theory is taught today using field extensions rather than by actually solving polynomials, students also learn to view a field extension as a vector space over the smaller field; both o... www.freeglossary.com /Galois_theory   (1201 words)

 MC449 Galois Theory Galois theory is one of the first examples of methods from one branch of Mathematics being applied to solve problems in an apparently completely different area. The main theorem of Galois theory is one of the most beautiful theorems in all of mathematics, and extensions and applications of Galois theory are the subject of major research activities in algebra, geometry and analysis. Appreciate the significance of the Galois group of a polynomial as a group of permutations of the roots. www.mcs.le.ac.uk /Modules/Modules01-02/node76.html   (641 words)

 Evariste Galois Galois tried to start his own school of mathematics, but got no students, so he joined the National Guard -- "If a carcass is needed to stir up the people, I will donate mine." Galois was jailed for supposedly threatening the King, but was found 'not guilty' by a jury. Galois took it violently and was disgusted with love, with himself, and with his girl." A few days later Galois encountered some of his political enemies and "an affair of honor," a duel, was arranged. Galois was shot in the intestines, and was taken to the hospital. scidiv.bcc.ctc.edu /Math/Galois.html   (653 words)

 Galois theory   (Site not responding. Last check: 2007-10-22) In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomial s. Further abstraction of Galois theory is achieved by the theory of Galois... Galois, Évariste (1811-1832) Galois theory, a branch of mathematics dealing with the general solution of equations, group theory, method of determining when a general equation could be solved by radicals, solved many long-standing unanswered questions. www.serebella.com /encyclopedia/article-Galois_theory.html   (2500 words)

 Galois theory   (Site not responding. Last check: 2007-10-22) In other words, the Galois theory is the study of solutions to polynomialsand how the different solutions are related to each other. In the modern approach, the setting is changed somewhat, in order to achieve a precise and more general definition: one startswith a field extension L/K and defines its Galois group as the group of all field automorphisms of L which keep all elements of K fixed. The notion of a solvable group in group theory allows us to determine whether or not a polynomial is solvable in the radicals, dependingon whether or not its Galois group has the property of solvability. www.therfcc.org /galois-theory-31875.html   (970 words)

 Category:Galois theory - Wikipedia, the free encyclopedia In mathematics, Galois theory is a branch of abstract algebra. At the most basic level, it uses permutation groups to describe how the various roots of a given polynomial equation are related to each other. This was the original point of view of Évariste Galois. en.wikipedia.org /wiki/Category:Galois_theory   (123 words)

 Algebraic homotopy, Galois theory and Descent   (Site not responding. Last check: 2007-10-22) To develop the Galois theory and Galois cohomology of higher dimensional algebraic, geometric and combinatorial structures inherent in Algebraic Homotopy and Algebraic Geometry. As one would expect, mixes of Galois theory and descent are being applied back in algebraic geometric settings with results on ringed topoi, whilst work by Brown continues to look at essential obstructions to extending local to global information measurable by monodromy groupoids generalising the fundamental group. The corresponding Galois theory and resulting algebraic homotopy is being studied at Bangor, not only for the potential application within equivariant homotopy and the theory of orbifolds, but also as a test bed for methods relating to Grothendieck's programme. www.informatics.bangor.ac.uk /public/research/mathematics/cathom/intas99.shtml   (1099 words)

 Galois theory Symmetries are usually expressed in terms of symmetry groups, and in fact the very notion of a group was invented by Evariste Galois to describe symmetries of roots. The Galois group of L over K is S, by a basic result of Artin. This is called the inverse Galois problem, and is usually posed for extensions of the rational number field Q. www.knowledgefun.com /book/g/ga/galois_theory.html   (1017 words)

 MA3D5 Galois Theory Content: Galois theory is the study of solutions of polynomial equations. In contrast, Ruffini, Abel and Galois discovered around 1800 that there is no such solution of the general quintic. E Artin, Galois Theory, University of Notre Dame. www.maths.warwick.ac.uk /undergrad/pydc/pink/pink-MA3D5.html   (339 words)

 Galois Theory   (Site not responding. Last check: 2007-10-22) Galois theory is one of the jewels of mathematics. This undergraduate text develops the basic results of Galois theory, with Historical Notes to explain how the concepts evolved and Mathematical Notes to highlight the many ideas encountered in the study of this marvelous subject. Section 13.2 discusses how to compute the Galois group of a quintic polynomial and in Example 13.2.13 mentions the problem of finding the roots of a quintic that is solvable by radicals. www.cs.amherst.edu /~dac/galois.html   (659 words)

 4H Galois Theory   (Site not responding. Last check: 2007-10-22) Galois was the James Dean of mathematics: he lived fast (arrested twice for revolutionary activities) and died young (in a pistol duel over a woman, aged 20). Of course, he also invented Galois Theory, and in doing so both solved one of the outstanding problems of contemporary mathematics and sowed the seeds of modern algebra. Abel prepared the ground for Galois: he showed that there is no general formula for solving the quintic, a result that Galois later refined. www.maths.gla.ac.uk /~tl/galois   (288 words)

 PlanetMath: fundamental theorem of Galois theory "fundamental theorem of Galois theory" is owned by djao. See Also: Galois-theoretic derivation of the cubic formula, Galois-theoretic derivation of the quartic formula, infinite Galois theory, Galois group This is version 3 of fundamental theorem of Galois theory, born on 2002-01-05, modified 2004-01-19. www.planetmath.org /encyclopedia/FundamentalTheoremOfGaloisTheory.html   (152 words)

 ABSTRACT ALGEBRA ON LINE: Galois Theory To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered permutations of the roots that leave the coefficient field fixed. The next lemma shows that in computing Galois groups it is enough to consider polynomials with integer coefficients. Then a powerful technique is to reduce the integer coefficients modulo a prime and consider the Galois group of the reduced equation over the field GF(p). www.math.niu.edu /~beachy/aaol/galois.html   (1898 words)

 GALOIS' THEORY OF ALGEBRAIC EQUATIONS Galois' Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field". A brief discussion on the fundamental theorems of modern Galois theory is included. www.worldscibooks.com /mathematics/4628.html   (481 words)

 Galois Theory Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields. The main topic of the course is undoubtedly Galois theory. Calculate the Galois group of the extension L:Q, where L is the splitting field of the minimal polynomial of the square root of a. math.berkeley.edu /~ribet/114   (1408 words)

 Further directions in Galois theory Galois theory is an important tool for studying the arithmetic of ``number fields'' (finite extensions of Q) and ``function fields'' (finite extensions of F Galois theory of Riemann surfaces: covering maps as field extensions, Galois groups and fundamental groups of punctured Riemann surfaces. Galois theory and algebraic geometry: Some of the same ideas in the algebraic setting, including varieties of degree >1. www.math.harvard.edu /~elkies/M250.01/galois_topix.html   (611 words)

 Algebraic Number Theory Archive   (Site not responding. Last check: 2007-10-22) ANT-0301: 18 Jun 2001, Octahedral Galois representations arising from Q-curves of degree 2, by Julio Fernández-González, Joan-Carles Lario, and Anna Rio. ANT-0185: 7 Jun 1999, An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z), by Avner Ash and Warren Sinnott. ANT-0115: 9 Jun 1998, The prime-to-adjoint principle and unobstructed Galois deformations in the Borel case, by Gebhard Boeckle and Ariane Mezard. front.math.ucdavis.edu /ANT   (12251 words)

 MAS316, Galois Theory Field theory: prime fields and characteristics, finite field extensions, simple extensions, principal element theorem, degree of an extension, product rule for degree, splitting fields, automorphisms of field extensions, embedding of one finite extension into another, separability, normal extensions, fundamental theorem of Galois theory. Field theory: prime fields and characteristic, finite field extensions, simple extensions, principal element theorem, degree of an extension, product rule for degree, splitting fields, automorphisms of field extensions, embedding of one field extension into another, separability, normal extensions, fundamental theorem of Galois theory. Applications: Insolubility of equations of degree greater than or equal to 5 by radicals, equivalence with insolubility of the Galois group, specific examples of insoluble equations over the rationals, ruler and compass constructions, symmetric polynomials (are generated by elementary symmetric polynomials). www.maths.qmw.ac.uk /undergraduate/modules/MAS316.html   (199 words)

 week201 Galois proof of the unsolvability of the quintic by radicals is just a more sophisticated variation on this theme. Then we define the "Galois group of K over k" to be the group of all automorphisms of K that act as the identity on k. Another point: there is no need for Galois theory to prove that duplication of the cube and trisection of an angle cannot be done by ruler and compass. math.ucr.edu /home/baez/week201.html   (5403 words)

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