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Topic: Abelian extensions

Related Topics

Abelian group

abelian categories

abelian varieties

free abelian group

finitely generated abelian groups

rank of an abelian group

abelian extensions

pre Abelian categories

category of abelian groups

Topological abelian group

arithmetic of abelian varieties

Abelian and tauberian theorems

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	Abelian extension - Wikipedia, the free encyclopedia
		In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian.
		In general a cyclotomic extension formed by adjoining roots of unity is abelian.
		The Kummer theory gives a complete description of the abelian extension case, and the Kronecker-Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.
	en.wikipedia.org /wiki/Abelian_extension (265 words)

	Stark's Conjecture in the Octahedral case (Site not responding. Last check: 2007-10-20)
		The classical case motivating the problem is that abelian extensions of Q can be formed by adjoining cyclotomic units, which are made using the exponential function.
		The smallest extension of Q with such a representation is of degree 48 with G conjugate to GL_2(F_3).
		This additional extension cannot be derived from an elliptic curve and yet the lattice of f's producing units is the same as for the other seven extensions.
	www.ma.utexas.edu /~kfogel/research.html (782 words)

	[No title]
		Cyclic extension L:K of number fields, Galois group generated by sigma, order g; g-module structure on the group of fractional ideals generated by all primes of L over P. Herbrand quotient of this group is 1/ef.
		A finite abelian extension of K in C corresponds to a closed subgroup of ideles containing principal ideles and having finite quotient.
		Summary of infinite class field theory: an abelian extension of K in C corresponds to a closed subgroup of ideles containing principal ideles and having totally disconnected quotient.
	cr.yp.to /2000-515/inclass.html (3147 words)

	Introduction to "Taming Wild Extensions" of Lindsay N. Childs
		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.
		The observation of {Ch87} for abelian extensions and {CM94} in general, was that for wild Galois extensions, if the associated order \frak A is a Hopf order in KG, then S is free of rank one over \frak A. Noether's theorem is the case where \frak A = RG.
		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)

	PlanetMath: Galois groups of finite abelian extensions of $\mathbb{Q}$
		The general case follows immediately from the above argument, the fundamental theorem of finite abelian groups, and a theorem regarding the Galois group of the compositum of two Galois extensions.
		Cross-references: Galois group of the compositum of two Galois extensions, subfield, normal subgroup, abelian, subgroup, divides, root of unity, prime, Dirichlet's theorem on primes in arithmetic progressions, cyclic, number fields, abelian group
		This is version 7 of Galois groups of finite abelian extensions of
	planetmath.org /encyclopedia/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ.html (175 words)

	PlanetMath: abelian number field
		Definition 1 An abelian number field is a number field
		The abelian number fields are classified by the Kronecker-Weber Theorem.
		This is version 2 of abelian number field, born on 2006-06-20, modified 2006-07-19.
	planetmath.org /encyclopedia/AbelianNumberField.html (78 words)

	Creation
		The most powerful way to create class fields or abelian extensions in Magma is to use the AbelianExtension function that enables the user to create the extension corresponding to some ideal group.
		The ray class group is returned as an abelian group A, together with a mapping between A and a set of representatives for the ray classes.
		The abelian extensions of Q are known to lie in some cyclotomic field.
	www.umich.edu /~gpcc/scs/magma/text661.htm (1839 words)

	Class field theory - Wikipedia, the free encyclopedia
		These days the term is generally used synonymously with the study of all the abelian extensions of algebraic number fields, or more generally of global fields; an abelian extension being a Galois extension with Galois group that is an abelian group.
		The point in general terms is to predict or construct the extensions of this type for a general number field K, in terms of the arithmetical properties of K itself.
		In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K; and associated to A a Galois group G which will be a pro-finite group, so a compact topological group, and also abelian.
	en.wikipedia.org /wiki/Class_field_theory (676 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)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-10-20)
		math.NT/0409352: 20 Sep 2004, Abelian surfaces of GL2-type as Jacobians of curves, by Josep Gonzalez, Jordi Guardia, Victor Rotger.
		ANT-0296: 8 Jun 2001, On the Iwasawa theory of p-adic Lie extensions, by Otmar Venjakob.
		ANT-0087: 1 Dec 1997, Degeneration of the l-adic Eisenstein symbol and of the elliptic polylog, by Annette Huber and Guido Kings.
	front.math.ucdavis.edu /ANT (12251 words)

	Introduction
		Class field theory is concerned with the classification of all abelian extensions of a given field.
		Abstractly, class field theory parametrizes abelian extensions in terms of abelian groups defined with respect to the base field.
		Class field theory classifies all abelian extensions of a given number field k in terms of quotients of ray class groups.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text755.htm (1077 words)

	Class Field Theory
		The ray class group is returned as an abelian group, together with a mapping between this abstract group and a set of representatives for the ray classes.
		The ultimate goal of class field theory was to be able to classify all abelian extensions of a given number field.
		The abelian extensions of Q are well known, all of them are contained in some cyclotomic field.
	www.math.niu.edu /help/math/magmahelp/text672.html (763 words)

	Cornelius Greither - Abstracts of recent work
		The first nontrivial case that avoids trouble with the prime 2 is the following: let K range over a family of imaginary sextic abelian extensions of Q, and study the distribution of the 3-parts of the minus class groups as G-modules, where G is the 3-part of Gal(K/Q).
		Abstract: We consider abelian extensions K of Q (the rationals) which are their own genus field and whose Galois group is the product of l cyclic groups of odd prime order p; we are interested in obtaining divisibilities of the class number of K by high powers of p.
		From work of Holland, and joint work of Holland and the author [31], it is known that for all abelian extensions L of Q, the invariant $\Omega(3,L/Q)$ lies in the so-called kernel group D(ZG) (a subgroup of Cl(ZG) which tends to be rather large).
	www1.informatik.unibw-muenchen.de /Greither/abstracts.html (1475 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		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.
		Seeking criteria for deciding if a given extension of local fields has an associated Hopf order in some Hopf algebra over which the extension is Hopf Galois (here the best recent work is that of N. Byott, at Exeter).
	math.albany.edu:8000 /~lc802/localgmt.html (613 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The Kronecker-Weber Theorem states that every abelian extension of the rationals is contained in a cyclotomic extension, which can be generated via the exponential function.
		The theory of complex multiplication uses elliptic functions to produce abelian extensions of complex quadratic fields.
		Furthermore, adjoining the w^th root of the Stark unit to K produces an abelian extension of k, where w is the number of roots of unity in K. In 2001, Stark realized that the conjectures should remain valid for any situation where the L-functions vanish at s=0.
	math.ucsd.edu /~erickson/research/synopsis.txt (619 words)

	Math 511B/512B
		denotes the maximal, abelian extension of F (which is an extension of F of infinite degree).
		But one of the nicest proofs of the theorem is based on the fact that every quadratic extension of Q is a subfield of some cyclotomic extension of Q.
		This result is a special case of the much more difficult "Kronecker-Weber" theorem which asserts that if K is any finite, abelian extension of Q, then K is a subfield of some cyclotomic extension of Q.
	www.math.washington.edu /Grads/Courses/1999-2000/511b.html (585 words)

	Class Field Theory
		In the number field case, all abelian extensions can be parametrized using more general class groups, in the case of global function fields, the same will be achieved using the divisor class group and extensions of it.
		Since the ray class field modulo m is always an infinite field extension containing the algebraic closure of the constant field, this returns Infinity.
		Provided F is abelian, this function will compute a divisor m and a sub group U of the ray class group modulo m sutch that F is isomorphic to the ray class field thus defined.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text779.htm (2785 words)

	A description of my work.
		The ergodic theory of systems with compact Lie group symmetry (in particular compact group extensions of hyperbolic systems) has undergone a resurgence in the last ten or so years as part of a programme to understand partially hyperbolic systems.
		In particular we are interested in the question of when the existence of a measurable solution to a cohomological equation posed on a group extension of an ergodic dynamical system implies the existence of a continuous solution.
		This question is crucial not only to the classification of the group extension but also to its mixing and stability properties.
	www.math.uh.edu /~nicol/mywork2.html (823 words)

	Dominions in metabelian groups
		We do have some interesting results, particularly for the metabelian groups (groups which are extensions of abelian or abelian; equivalently, groups for which the commutator subgroup is abelian; also equivalently, solvable of length at most two).
		Part of the reason this may seem surprising is that this means that H, which is a direct summand of rank three of that countably generated free abelian group actually dominates that entire abelian subgroup and more.
		A similar argument works if we consider the variety of all extensions of abelian groups of exponent n by abelian groups of exponent m (a subvariety of the variety of all metabelian groups).
	math.berkeley.edu /~magidin/research/metab.html (487 words)

	Shimura, G.: Introduction to Arithmetic Theory of Automorphic Functions.
		At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed.
		The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.
		Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves
	press.princeton.edu /titles/5530.html (207 words)

	Math JS Milne Preprints
		Prove the full conjecture of Birch and Swinnerton-Dyer in the case of a constant abelian variety over a global field of prime characteristic; in particular, give the first examples of nonzero abelian varieties whose Tate-Shafarevich groups are known to be finite.
		For an abelian variety A and its dual B over a local field of prime characteristic, prove that A(K) is dual to the Weil-Chatelet group of B.
		Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a good category of abelian motives over the algebraic closure of a finite field and a reduction functor to it from the category of CM-motives.
	www.jmilne.org /math/Preprints/index.html (2485 words)

	Kummer theory on extensions of abelian varieties by tori, Kenneth A. Ribet (Site not responding. Last check: 2007-10-20)
		Kummer theory on extensions of abelian varieties by tori, Kenneth A. Ribet
		Kummer theory on extensions of abelian varieties by tori
		Zarkhin, Endomorphisms of abelian varieties and points of finite order in characteristic $p$, Mathematical Notes of the Academy of Sciences of the USSR 21 (1977), 415–419, Matematicheskie Zametki 21 (1977), 737–744.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1077313720 (439 words)

	Math Events (Site not responding. Last check: 2007-10-20)
		Specific topics to be covered are the zeta-function of a curve, the Hasse-Weil theorem, Serre´s improvement of the Hasse-Weil bound, explicit formulas and the Drinfeld-Vladut bound, an asymptotically optimal tower of curves with wild ramification, several constructions of curves with many rational points (following v.d.Geer-v.d.Vlugt, Garcia et al.).
		The following topics will be covered; Galois extensions of algebraic function fields, ramification of places in Galois extensions, ramification groups, different exponents in Galois extensions, Hasse´s theory of abelian extensions, computation of the genus in abelian (Kummer and Artin-Schreier type) extensions, Abhyankar´s lemma.
		Financial aid, covering accommodation, is available especially to graduate students with priority given to those participants from Turkish universities and research institutes.
	www.sabanciuniv.edu /math_events/curves.html (290 words)

	Class Field Theory
		In the number field case, all abelian extensions can be parameterized using more general class groups, in the case of global function fields, the same will be achieved using the divisor class group and extensions of it.
		Let m be an effective divisor and U be a subgroup of the ray class group, (see RayClassGroup), modulo m such that the quotient Cl_m/U is finite.
		Provided F is abelian, this function will compute a divisor m and a sub group U of the ray class group modulo m such that F is isomorphic to the ray class field thus defined.
	www.math.lsu.edu /magma/text715.htm (2760 words)

	Table of contents for Library of Congress control number 2002034849
		The Cogalois group of a quadratic extension 83 3.4.
		The Kneser group of a G-Cogalois extension 104 4.5.
		Kummer extensions with few roots of unity 180 7.4.
	www.loc.gov /catdir/toc/fy038/2002034849.html (292 words)

	Math 254B Syllabus (Site not responding. Last check: 2007-10-20)
		Math 254B, as the continuation of Math 254A, is a second-semester graduate course in algebraic number theory.
		Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century.
		Abelian extensions of the rationals: the Kronecker-Weber theorem (Milne, I.4; Washington, Chapter 14)
	www-math.mit.edu /~kedlaya/Math254B/syllabus.html (547 words)

	Mathematics Colloquium #1 (Site not responding. Last check: 2007-10-20)
		After observing that additive Galois structure is dependent upon Hilbert's ramification filtration, we will specialize to fully ramified p-extensions of local number fields (the totally wild extensions).
		Here Hilbert's ramification filtration does not provide us with a rich enough source of invariants (to determine additive Galois structure), and we are forced to look for a richer source.
		The result is an interesting generalization of the group and a new ramification filtration.
	www.unomaha.edu /~wwwmath/OurArchive/colloquium/Fall2004/coll2.html (146 words)

	Abelian Extensions
		Several new functions that exploit the Galois-module structure of the ray-class-groups.
		Functions to convert number fields into abelian extensions.
		A ``guess'' can be computed without the knowledge of defining equations.
	www.umich.edu /~gpcc/scs/magma/rel/node27.htm (30 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		In several regards, the concept of a Drinfel'd module is analogous to the concept of an elliptic curve (or more generally, of an irreducible Abelian variety), with which it shares many features.
		On the other hand, since the mechanism of Drinfel'd modules is smoother and in some respects simpler than that of Abelian varieties, some results involving Drinfel'd modules over global function fields
		Adjoining torsion points of rank-one Drinfel'd modules results in Abelian extensions of the base field.
	eom.springer.de /D/d120270.htm (1069 words)

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