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Topic: Galois extension

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Field extension

Galois group

Galois theory

Langlands program

Group isomorphism

Lie group

Normal morphism

Evariste Galois

Robert Langlands

Morphism

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	Galois group - Wikipedia, the free encyclopedia
		Suppose that E is an extension of the field F.
		The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.
		The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the subgroups of the Galois group correspond to the intermediate fields of the field extension.
	en.wikipedia.org /wiki/Galois_group (384 words)

	Galois extension - Wikipedia, the free encyclopedia
		In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois.
		The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
		The extension E/F is Galois if the field fixed by the automorphism group Aut(E/F) is precisely the base field F.
	en.wikipedia.org /wiki/Galois_extension (229 words)

	Galois Implies Separable (Site not responding. Last check: 2007-10-20)
		An algebraic galois extension f over k is separable, and normal, and acts as a splitting field for a set of separable polynomials.
		In summary, the extension f/k is the splitting field for a set of separable polynomials iff f is galois and algebraic.
		An infinite algebraic extension is galois iff it splits a set of separable polynomials, iff it is the union of finite galois extensions.
	www.mathreference.com /fld-sep,galois.html (500 words)

	Math Forum - Ask Dr. Math
		In this case, the extensions are: Q < Q(sqrt(6)) < Q(sqrt(1+sqrt(6))) Q < Q(sqrt(6)) < Q(sqrt(1-sqrt(6))) and the latter two extensions are not the same, since the first one is real and the second one is not (in this case, the Galois group is D4, the dihedral group of order 8).
		This shows that all the extensions in the chain [2] are of degree 1 or 2, and therefore the degree of splitting field is a power of 2.
		For the purpose of computing Galois groups, however, this is irrelevant--all operations take place in the splitting field of f, and we are not restricted to the real part of it.
	mathforum.org /library/drmath/view/66643.html (712 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		MU is an S[BU]-Hopf-Galois extension of commutative S-algebras, with coaction fi :MU !
		Thus the E-local Galois conditions, that G is stably dualizable and the maps* * i and h are weak equivalences, are invariant under changes in A, B or G that amou* *nt to E-local weak equivalences of A, B and S[G].
		Galois extensions of commutative rings are always faithfully flat, and it wi* *ll be convenient to consider the corresponding property for structured ring spectra.
	www.math.purdue.edu /research/atopology/Rognes/galois.txt (18535 words)

	PlanetMath: the compositum of a Galois extension and another extension is Galois
		"the compositum of a Galois extension and another extension is Galois" is owned by alozano.
		See Also: fundamental theorem of Galois theory, Galois extension, example of normal extension, class number divisibility in extensions, Galois group of the compositum of two Galois extensions, extensions without unramified subextensions and class number divisibility
		This is version 3 of the compositum of a Galois extension and another extension is Galois, born on 2005-02-21, modified 2005-03-10.
	www.planetmath.org /encyclopedia/CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois.html (153 words)

	Galois Groups (Site not responding. Last check: 2007-10-20)
		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).
		Everything in the extension is real, and the root cannot be mapped onto a complex cube root of 2, so the galois group is trivial, even though the dimension of the extension is 3.
	www.mathreference.com /fld-gal,gg.html (448 words)

	Science Fair Projects - Category:Galois theory
		In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials.
		In other words, the Galois theory is the study of solutions to polynomials and how the different solutions are related to each other.
		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.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Category:Galois_theory (245 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)

	.:: Galois Field Arithmetic Library ::.
		The branch in mathematics known as Galois theory (pronounced as "gal-wah") which is based on abstract algebra was discovered by a young brilliant french mathematician known as Evariste Galois.
		Galois theory is used to describe and generalize results seen in these fields, for example the AES algorithm can be represented with just a few lines of mathematics using Galois theory and some other related abstract algebra.
		Galois fields are setup by intially defining the size of the field which means how many elements will exist within the field, and also the values those elements will posses.
	www.partow.net /projects/galois/index.html (737 words)

	Galois extension
		In mathematics, a Galois extension is a field extension that has a Galois group.
		A fundamental result of Galois theory characterises these extensions: a finite extension of fields L/K is a Galois extension if and only if it is both a normal extension and a separable extension.
		It states, in more concrete terms, that L is built up from K as a compositum of a number of splitting fields of separable polynomials.
	www.starrepublic.org /encyclopedia/wikipedia/g/ga/galois_extension.html (137 words)

	Galois group
		Suppose E is an extension of the field F, and consider the set of all field automorphisms of E which fix F pointwise.
		This function is monotone decreasing and its inverse is given by the Galois group of E/E
		induces an isomorphism between the group G/H and the Galois group of the extension E
	www.ebroadcast.com.au /lookup/encyclopedia/ga/Galois_group.html (249 words)

	Galois Theory Glossary
		An extension of fields L/K (this notation does not denote any sort of quotient) is a ring homomorphism K --> L. Such a homomorphism has to be injective, so that K is isomorphic to a subfield of L. It is often convenient to identify K with this subfield.
		A Galois extension is a normal separably algebraic extension of finite degree.
		Let L/K be an extension of fields, and let a in L be algebraic over K. The minimal polynomial of a over K is the unique monic polynomial f(t) in K[t] of least degree such that f(a) = 0.
	www.wra1th.plus.com /Galois/gloss.html (892 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		For Galois extensions of local fields which are wildly ramified (i.e.
		Leopoldt showed that if K = Q, the rational numbers, and L is any abelian extension of Q (wild or tame), then S is free as an A-module, thereby generalizing the 19th century Hilbert-Speiser theorem for tame abelian extensions of Q. Two basic results created an interesting new approach to wild extensions.
		Algebra, 1996) that a Galois extension L/K with Galois group G has a unique Hopf Galois structure iff G is cyclic of order n where n and (Euler's phi function)(n) are coprime.
	math.albany.edu:8000 /~lc802/localgmt.html (613 words)

	Conley: Galois Theory, Summer 2000
		Let $K$ be a field of characteristic~0, and let $L$ be a Galois extension of $K$ whose Galois group is $C_p$, for some prime~$p$.
		Prove that the Galois group of $F$ over $\Bbb Z_p$ is cyclic, and generated by $\sigma$.
		Use the Galois correspondence to describe all subfields of $F$.
	www.math.unt.edu /~conley/Galois0008.htm (591 words)

	Gatorsports.com :: 100 years of Gator Football (Site not responding. Last check: 2007-10-20)
		In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group G.
		Equivalently, for a Galois group G and a group ring R G" class="external">[1] of G with respect to some ring R, a Galois module is some R G" class="external">[2]-module M.
		Certainly G can either be an infinite profinite Galois group, or a finite one for a finite Galois extension L/K of fields.
	www.gatorsports.com /apps/pbcs.dll/section?template=wiki&text=Galois_module (687 words)

	Galois extension - Encyclopedia, History, Geography and Biography
		Galois extension - Encyclopedia, History, Geography and Biography
		[E:F] = Aut(E/F); that is, the degree of the field extension is equal to the order of the automorphism group of E/F. Category
		Galois extension, Characterization of Galois extensions and Galois theory.
	www.arikah.com /encyclopedia/Galois_extension (250 words)

	Galois theory - AoPSWiki
		Galois theory is an important tool for the study of fields.
		The primary objects of study in Galois theory are automorphisms of fields.
		This group is called the Galois group of L / K and is denoted Gal(L / K).
	www.artofproblemsolving.com /Wiki/index.php/Galois_theory (304 words)

	proof concerning galois extension
		There is already a widely accepted notation for this group, "Aut(E/F)", the automorphisms of E over F, that does not imply (as his notation does) that the extension is Galois in the first place.
		It seems to me that talking about "the Galois group" of a non-Galois extension is pretty much as poor a choice of notation as you can get here.
		So you are trying to show that a finite extension E/F is Galois if and only if the fixed field of Aut(E/F) is F. Your proof that if the extension is Galois then the fixed field is F seems to be correct.
	www.forum-one.org /new-6376011-4346.html (1330 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.
		An algebraic extension field F of K is called separable over K if the minimal polynomial of each element of F is separable.
		Let K be a field of characteristic zero, and let E be a radical extension of K. Then there exists an extension F of E that is a normal radical extension of K. Theorem.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	ADFS::HD4.$.Work.courses.98-99.Galois.Notes.N4
		A Galois extension is a finite normal separable extension.
		It follows from this that intermediate extensions are conjugate if and only if the corresponding subgroups of the Galois group are conjugate.
		is not a Galois extension unless it is normal, and this happens precisely when it equals all its conjugates.
	www.wra1th.plus.com /Galois/N4.html (278 words)

	[No title]
		The Galois group G of [S5:Q] is of order 48, and is an extension of the permutation module M for Sym(3) with Sym(3).
		In that case L_1 is Galois and you do the same trick, >at each stage, adjoin not merely the square root of an element, but also >the square roots of each of its conjugates to get a Galois extension at >each stage.
		1, F_i is a galois extension of F_{i-1} of degree p (hence by a pth root).
	www.math.niu.edu /~rusin/known-math/01_incoming/2tower (2194 words)

	Encyclopedia (Site not responding. Last check: 2007-10-20)
		The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory.
		If F is Q (the field of rational numbers), and E is Q(√2), the field obtained from Q by adjoining √2, then the Galois group again has two elements: the identity automorphism, and the automorphism which exchanges √2 and −√2.
		This is because it is not a normal extension, since the other two cube roots of 2, being complex numbers, are not contained in Q(α).
	www.wikiworld.biz /galois_group (354 words)

	Generic Galois Extensions -- Saltman 77 (3): 1250 -- Proceedings of the National Academy of Sciences
		We define the notion of a generic Galois extension with group G over a field F. Let R be a commutative ring of the form F[x
		Assume K/L is a Galois extension of fields with group G and such that L\supseteq F. Then there is an F algebra map f:R
		L. We construct generic Galois extensions for certain G and F. We show such extensions are related to Noether's problem and the Grunwald-Wang theorem.
	www.pnas.org /cgi/content/abstract/77/3/1250 (207 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		Just as class field theory for unramified Abelian extensions can be explained in terms of the divisor class group and its subgroups, so can arbitrary Abelian extensions be characterized by means of ray class groups with respect to suitable modules (see Algebraic number theory).
		There are also generalizations of class field theory to the case of infinite Galois extensions [4].
		, is contained in an extension generated by the torsion points of an elliptic curve with complex multiplication.
	eom.springer.de /c/c022370.htm (975 words)

	► » proof concerning galois extension (Site not responding. Last check: 2007-10-20)
		notation does) that the extension is Galois in the first place.
		extension is pretty much as poor a choice of notation as you can get
		extension E/F: Gal(E/F) = { phi \in Aut(E) : phi(x)=x for all x\in F }
	www.science-chat.org /proof-concerning-galois-extension-6376011.html (807 words)

	Galois group - the free encyclopedia (Site not responding. Last check: 2007-10-20)
		If there are no elements of E \ F which are fixed by all members of G, then the extension E/F is called a Galois extension, and G is the Galois group of the extension and is usually denoted Gal(E/F).
		This can be explained in the terms that the other two cube roots of 2 are complex numbers; in other words E is not a
		isomorphism between the group G/H and the Galois group of the extension E
	www.free-web-encyclopedia.com /default.asp?t=Galois_group (337 words)

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