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# Topic: Galois group

###### In the News (Sun 16 Jun 19)

 Learn more about Galois theory in the online encyclopedia.   (Site not responding. Last check: 2007-11-05) Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. and the Galois group is isomorphic to the Klein four-group. The notion of a solvable group in group theory allows us to determine whether or not a polynomial is solvable in the radicals, depending on whether or not its Galois group has the property of solvability. www.onlineencyclopedia.org /g/ga/galois_theory.html   (753 words)

 Galois group - Wikipedia, the free encyclopedia In mathematics, a Galois group is a group associated with a certain type of field extension. The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory. If F is the field R of real numbers, and E is the field C of complex numbers, then the Galois group has two elements, namely the identity automorphism and the complex conjugation automorphism. en.wikipedia.org /wiki/Galois_group   (394 words)

 Galois theory - Wikipedia, the free encyclopedia In mathematics, Galois theory is a branch of abstract algebra. 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   (1577 words)

 Kids.net.au - Encyclopedia Galois theory - 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. If a factor group in the composition series is cyclic of order n, then the corresponding field extension is a radical extension, and the elements of L can then be expressed using the nth root of some element of K. If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q). www.kids.net.au /encyclopedia-wiki/ga/Galois_theory   (686 words)

 Galois theory   (Site not responding. Last check: 2007-11-05) In mathematics, 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 very notion of a group was invented by Evariste Galois to describe symmetries of roots. In the modern approach, the setting 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.sciencedaily.com /encyclopedia/galois_theory   (1068 words)

 Galois   (Site not responding. Last check: 2007-11-05) Galois' father was an important man in the community and in 1815 he was elected mayor of Bourg-la-Reine. Galois was invited by Poisson to submit a third version of his memoir on equation to the Academy and he did so on 17 January. Galois was wounded in the duel and was abandoned by d'Herbinville and his own seconds and found by a peasant. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Galois.html   (1959 words)

 Galois representations and elliptic curves   (Site not responding. Last check: 2007-11-05) Galois' brilliant insight was that one can know essentially "everything" there is to know about the roots of polynomial equations by considering a new object, a group, namely the group of all "reasonable" permuations of those roots. The Galois group is a way of encoding all available information about the relationships of the roots of polynomials with coefficients in the base field that factor completely in the extension field. F is Galois if E is the field obtained by adjoining to F all roots of some irreducible polynomial with coefficients in F. The Galois group of E over F, Gal(E/F), is the group of automorphisms of E that leave F fixed (i. www.mbay.net /~cgd/flt/flt07.htm   (3077 words)

 The Galois group of a polynomial   (Site not responding. Last check: 2007-11-05) A method for deciding whether the Galois group is abelian... Determining the Galois group of a rational polynomial... Computing the Galois group of a polynomial using linear differential equations... www.scienceoxygen.com /math/294.html   (91 words)

 The Galois Group of a Polynomial   (Site not responding. Last check: 2007-11-05) The Galois group of the polynomial is then the group G of automorphisms of E that leave F fixed. This, therefore, is the Galois group of the preceding polynomial relative to the coefficient field Q. One of the most important applications of Galois theory (indeed, the reason it was invented) is to provide the criterion for deciding when a polynomial is solvable by means of rational operations and root extractions. From group theory it can be shown that this is possible for the fully symmetric permutation groups of two, three, or four entities (of orders 2, 6, and 24 respectively), but not for the fully symmetric group of five (or more) entities. www.mcs.drexel.edu /~rboyer/courses/math534/galois_grp_math_world   (2033 words)

 The Hindu : Brief life of a mathematician Galois was aware that he was fighting a superior adversary, and that most likely he would be killed in the duel. Galois in 1831 was the first to really understand that the algebraic solvability of a polynomial equation was intimately related to the group structure of certain permutations associated with the equation. The understanding of the general group concept, and the realisation that it was the basis of Galois' work, came only in the second half of the 19th century. www.hinduonnet.com /2000/12/31/stories/13310469.htm   (2892 words)

 Introduction to "Taming Wild Extensions" of Lindsay N. Childs Galois module theory is the branch of algebraic number theory which studies rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group. Since wild extensions include all ramified Galois extensions of a local field K containing \Bbb Q_p where the Galois group is a p-group, this was a substantial omission. We then give Byott's classification of Galois extensions for which the classical Galois structure is the unique Hopf Galois structure, and survey results on the number of Hopf Galois structures on Galois extensions with Galois group G for various G, including cyclic p-groups {Ko98}, and symmetric, alternating and simple groups {CC99}. math.albany.edu:8000 /~lc802/mono.html   (1829 words)

 The Group Concept   (Site not responding. Last check: 2007-11-05) Any given group is isomorph to a given abstract group and its structure is characterized by all the abstract subgroups of this abstract group. Group theory is the study of those properties of groups which are preserved under isomorphism. This abstract group being the lowest-order non-Abelian group. www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node159.html   (1396 words)

 PlanetMath: Galois group The group operation is given by composition: for two automorphisms is defined to be the Galois group of the splitting field of This is version 4 of Galois group, born on 2002-01-05, modified 2004-07-28. planetmath.org /encyclopedia/GaloisGroup.html   (78 words)

 Galois Groups   (Site not responding. Last check: 2007-11-05) The galois group of f over k is the set of automorphisms of f that fix k. Using the terminology of group theory, the galois group acts transitively on the roots of p(x). Every automorphism on f/e also fixes k, hence the galois group of f over e is a subgroup of the galois group of f over k. www.mathreference.com /fld-gal,gg.html   (448 words)

 [No title] Discuss sufficient conditions on a polynomial of degree 5 to have Galois group S_5 (two nonreal roots.) What are necessary and sufficient conditions for the Galois group to have an order a multiple of 3. Give a condition on the Galois group that is implied by the irreducibility of the polynomial. Suppose you have a finite p-group and you have a representation of this group on a finite dimensional vector space over a finite field of characteristic p. www.math.princeton.edu /graduate/generals/algebra.txt   (924 words)

 UF Mathematics: About Group Theory   (Site not responding. Last check: 2007-11-05) Research in group theory at the University of Florida is carried out by mathematicians in Algebra Research Group. When Galois investigated the solvability of polynomial equations in terms of radicals, he studied certain types of groups associated with such equations which are called Galois groups today. The Inverse Galois Problem concerns the question whether for any given group there exists a polynomial equation such that the Galois group associated with the solution of this equation is the given group. www.math.ufl.edu /dept_news_events/honors/group_theory.html   (530 words)

 Determining the Galois Group of a Polynomial Now, by a theorem of Cebotarev we know the proportion of all primes relative to which f(x) splits is asymptotic to 1/order(G) where G is the galois group of f. Clearly the order of the group is not 720 (which it would have to be if the Galois group was S_6), so it must be a proper subgroup. The point is that by just playing around with f(x) in various F_p[x] the Galois group of the polynomial quickly becomes very apparent, even without slogging through a rigorous demonstration. www.mathpages.com /home/kmath485.htm   (326 words)

 Galois' commentators For instance, Kronecker was first to describe the Galois group not in terms of permutations on the roots of an equation, but as a group of automorphisms of the coefficient field with adjoined quantities. Weber presented Galois Theory in terms of group theory and field theory, making very few references to equations, so that the theory could also be applied to other areas than the solvability of equations. Today's formulations (see [6] or [7]) of the Fundamental Theorem of Galois Theory are equivalent to Artin's; their aim is to reveal the parallel structure of the extension field and its automorphism group. www-groups.dcs.st-and.ac.uk /~history/Projects/Brunk/Chapters/Ch3.html   (2035 words)

 The Galois Group of a Polynomial If f/k is a field extension, its galois group is the group of automorphisms of f that fix k. If p(x) is an irreducible polynomial with coefficients in k, it also has a galois group, namely the galois group of its splitting field. Since the size of the orbit, times the size of the stabilizing subgroup, gives the size of g, the order of the galois group is divisible by n, where n is the number of roots in p(x). www.mathreference.com /fld-slv,gpx.html   (658 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)

 Research Interests of Lindsay N. Childs Noether's theorem is the starting point for an extensive global theory which, for a tamely ramified Galois extension L/K of algebraic number fields, relates the class of the ring of integers S of L in the locally free class group Cl(RG) (or Cl(ZG)) to analytic invariants associated with L-functions. For wild extensions, one approach is Galois module theory is to replace the group ring RG by a larger order in KG, the associated order of S in KG, namely, the set of elements in KG which map S into itself (not just into L). Bondarko's paper looks at Hopf orders in group rings of abelian groups and shows, under some restrictions, that a Hopf order is realizable iff the Hopf order represents the kernel of an isogeny of dimension one formal groups (and is then necessarily monogenic). nyjm.albany.edu:8000 /~lc802/research.html   (876 words)

 Galois group and action The elements of the Galois group of A over are obtained by sending a to one of its conjugates. Thus the action on A of such an element of the Galois group is determined by the polynomial (with rational coefficients) expressing the conjugate in a. Lastly, we find the Galois group G using the intrinsic function once more, but in terms of the degree-6 defining polynomial of the splitting field. magma.maths.usyd.edu.au /magma/Examples/node63.html   (533 words)

 Amazon.com: Books: Inverse Galois Theory (Springer Monographs in Mathematics)   (Site not responding. Last check: 2007-11-05) Inverse Galois Theory is concerned with the question of which finite groups occur as Galois Groups over a given field. In particular, this includes the question of the structure and the representations of the absolute Galois group of K and also the question about its finite epimorphic images, the so-called inverse problem of Galois theory. The idea of deducing the realizability of a finite group as Galois group over Q(t) from the existence of rigid systems of generators as far as we know first appeared implicitly in the appendix to the dissertation of Shih (1974). www.amazon.com /exec/obidos/tg/detail/-/3540628908?v=glance   (627 words)

 [No title]   (Site not responding. Last check: 2007-11-05) Another point of departure is the theory of invariants of ordinary representations of finite groups--the algebra of invariants characterizes the group (which is a permutation subgroup of S_n, and hence also a linear rep, and hence has invariants=resolvents.) There was a nice survey in the Bullitin back in '79 or thereabouts on invariants of groups. Subject: Re: Galois group question > the way I always remembered it was that getting the galois group to be > less than s_n required some condition on the coefficients, e.g. the > galois group is contained in a_n iff the discriminant is a square. www.math.niu.edu /~rusin/known-math/95/galois   (742 words)

 Relative resolvents and partition tables in Galois group computations - Colin (ResearchIndex)   (Site not responding. Last check: 2007-11-05) The aim of the article is to lower the degrees of the resolvents to factorize over the ground field. For instance, the degrees of the resolvents to factorize in order to identify the Galois group of an irreducible polynomial of... 9 Computation of the Galois groups of the resolvent factors fo.. citeseer.ist.psu.edu /colin97relative.html   (514 words)

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