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	Ideal class group - Wikipedia, the free encyclopedia
		The first ideal class groups encountered in mathematics were part of the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms.
		This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R.
		Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group.
	en.wikipedia.org /wiki/Ideal_class_group (1238 words)

	Fractional ideal - Wikipedia, the free encyclopedia
		A fractional ideal of R is a nonzero finitely generated R-submodule of K.
		Fractional ideals can be multiplied in a natural manner, and it can be shown that in a Dedekind domain, the fractional ideals form an abelian multiplicative group with R as identity element.
		The quotient group of fractional ideals divided by principal fractional ideals is isomorphic to the ideal class group of R.
	en.wikipedia.org /wiki/Fractional_ideal (425 words)

	Ideal class group: Definition and Links by Encyclopedian.com - All about Ideal class group (Site not responding. Last check: 2007-11-07)
		We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem.
		The quotient group of fractional ideals divided by principal fractional ideals is the ideal class group of R; it is isomorphic to the one defined above.
		Class field theory is a more advanced branch of number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group.
	www.encyclopedian.com /id/Ideal-class-group.html (1396 words)

	PlanetMath: ideal classes form an abelian group (Site not responding. Last check: 2007-11-07)
		The ideal class group is one of the principal objects of algebraic number theory.
		"ideal classes form an abelian group" is owned by mathcam.
		This is version 7 of ideal classes form an abelian group, born on 2002-07-02, modified 2004-03-05.
	planetmath.org /encyclopedia/ClassNumber2.html (137 words)

	Ideal Class Groups (Site not responding. Last check: 2007-11-07)
		To determine the class group or class number correctly one has to make sure that all ideals having norm smaller than the Minkowski bound or smaller than the Bach bound, if one assumes the generalized Riemann hypothesis, are taken into consideration, and that the final stage, which may be time consuming, is properly executed.
		Ideals in Magma are discussed in Ideals and Quotients.
		An integral upper bound for norms of generators of the ideal class group for K or O assuming the generalized Riemann hypothesis.
	www.math.niu.edu /help/math/magmahelp/text667.html (921 words)

	PlanetMath: ideal class (Site not responding. Last check: 2007-11-07)
		By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of
		See Also: existence of Hilbert class field, fractional ideal, number field, unramified extensions and class number divisibility, class number divisibility in extensions, push-down theorem on class numbers, Minkowski's constant, extensions without unramified subextensions and class number divisibility, class number divisibility in
		This is version 18 of ideal class, born on 2002-04-23, modified 2003-09-22.
	planetmath.org /encyclopedia/IdealClass.html (187 words)

	Renate Scheidler - Research Interests
		A fractional ideal is simply an ideal in the ordinary sense with a "denominator" which in our context is a nonzero integer (in the number field setting) or a polynomial (in the function field case).
		The factor group of the former group by the latter is the ideal class group of K.
		The group of units O* of the maximal order O is an infinite Abelian group whose rank is the unit rank of K.
	www.math.ucalgary.ca /~rscheidl/research.html (1369 words)

	Class field theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Class field theory is a major branch of (Click link for more info and facts about algebraic number theory) algebraic number theory, including most of the central results that were proved in the period about 1900-1950.
		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.
		The class field theory project included the 'higher reciprocity laws' (cubic reciprocity and so on), but is not limited to that one, classical line of generalisation.
	www.absoluteastronomy.com /encyclopedia/c/cl/class_field_theory.htm (389 words)

	quadratic_order (Site not responding. Last check: 2007-11-07)
		elementary divisors of the ideal class group (
		The regulator is defined as the natural logarithm of the fundamental unit and the class number is the order of the group of ideal equivalence classes.
		We denote the regulator by R, the class group by CL and the class number by h.
	www.math.psu.edu /local_doc/LiDIA/node86.html (1935 words)

	Global Function Fields (Site not responding. Last check: 2007-11-07)
		The map from the multiplicative group of the field of fractions of O to the group of fractional ideals of O where O is a `finite' maximal order.
		The divisor class group of F/k as an Abelian group, a map of representatives from the class group to the divisor group and the homomorphism from the divisor group onto the divisor class group.
		The ideal class group of the `finite' maximal order O as an Abelian group, a map of representatives from the ideal class group to the group of fractional ideals and the homomorphism from the group of fractional ideals onto the ideal class group.
	magma.maths.usyd.edu.au /magma/htmlhelp/text708.htm (801 words)

	Ideal Class Groups (Site not responding. Last check: 2007-11-07)
		Using these relations, a generating set for the ideal class group is derived (via matrix echelonization), and in the final step it is verified that the correct orders for the generators has been found.
		To determine the class group (or number) correctly one has to make sure that all ideals having norm smaller than the Minkowski bound (or smaller than the Bach bound if one assumes the generalized Riemann hypotheses) are taken into consideration, and that the final stage (which may be time consuming) is properly executed.
		The ray class group is returned as an abelian group, together with a bijection between this abstract group and a set of representatives for the ray classes.
	www.dtr.isy.liu.se /Magma/text441.html (927 words)

	Class Group (Site not responding. Last check: 2007-11-07)
		The class group of a pid is trivial.
		Conversely, an ideal that is not principle becomes a nontrivial element in the class group.
		Two ideals, or fractional ideals, represent the same element in the class group if their quotient is principle.
	www.mathreference.com /id-dd,cgrp.html (330 words)

	Connections between Cubic and Dual Quadratic Fields
		class field theory [2] (the ARTIN correspondence between subgroups of the 3-elementary ideal class group of k and unramified cyclic cubic extensions of k, resp.
		KUMMER theory [3] (the KUMMER correspondence between subgroups of the 3-radicand group of k' and cyclic cubic super fields of k, resp.
		the principal ideal cubes associated with generating ideal classes, which are joined by the fundamental unit e of k' as a further principal ideal cube.
	www.algebra.at /mirror.htm (801 words)

	Open Questions: Algebraic Number Theory
		Class field theory has a reputation for being a very difficult subject, and it is. There is a fair bit of abstract conceptual machinery involved even to explain many of the results, and (of course) much more to prove the results.
		On the other hand, class field theory provides a complete answer of a different kind which specifies a 1-to-1 correspondence between abelian extensions of a number field and certain generalized ideal class groups which are defined entirely from the arithmetic properties of the base field.
		As far as ramification is concerned, the class field K is the maximal abelian extension of k which is unramified except for primes that divide the conductor F. This is an alternative definition of K as the class field of k, so this too is a direct generalization of Hilbert's results.
	www.openquestions.com /oq-ma018.htm (19624 words)

	quadratic_ideal (Site not responding. Last check: 2007-11-07)
		It supports basic operations like ideal multiplication as well as more complex operations such as computing the order of an equivalence class in the class group.
		This class is meant to be used for general computations with quadratic ideals.
		The ideal is assumed to belong to the most recently accessed quadratic order.
	www.math.psu.edu /local_doc/LiDIA/node89.html (1928 words)

	Class Field Theory (Site not responding. Last check: 2007-11-07)
		The classical approach to class field theory, which is well suited for computations, is based on ideal groups which are generalisations of the ideal class group.
		In this case, the generators of the principal ideals have to take positive values at the places indicated by the sequence T. T must be an ascending sequence of positive integers with length and elements at most the number of real embeddings.
		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.
	www.mat.niu.edu /help/math/magmahelp/text672.html (763 words)

	On the unit groups and the ideal class groups of certain cubic number fields (Site not responding. Last check: 2007-11-07)
		On the unit groups and the ideal class groups of certain cubic number fields
		We consider the unit group of ${\bf Q}(\theta)$.
		And we consider the $3$-class group of ${\bf Q}(\theta)$.
	www.ma.kagu.sut.ac.jp /~sutjmath/39/no2/Yoshida.html (91 words)

	Structure Operations
		The class group of an order O or the maximal order of the quadratic field K, as an abelian group.
		The class number of O or the order O or the maximal order of the quadratic field K. The parameter Al may be supplied to select the method used to calculate the class number.
		The rank of the free part of the unit group of the order O or the maximal order of the quadratic field K, which equals 1 for real quadratic fields and 0 for imagnary quadratic fields.
	www.math.wisc.edu /help/magma/text436.html (720 words)

	7.2 Naive computation of the class group (Site not responding. Last check: 2007-11-07)
		As shown earlier, each element of the class group of the order R is represented by an invertible ideal J with
		Thus the involution acts on the class group by group inversion (which is a group homomorphism for abelian groups!).
		Once we find a group that is the correct range then there are techniques to verify that there are no more relations to be considered.
	www.imsc.ernet.in /~kapil/crypto/notes/node36.html (793 words)

	Ideal Arithmetic and Infrastructure in Purely Cubic Function Fields - Scheidler (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: This paper investigates the arithmetic of fractional ideals of a purely cubic function eld and the infrastructure of the principal ideal class when the eld has unit rank one.
		We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least...
		The algorithm was originally used for generating the entirety of reduced fractional principal ideals and thus nding the fundamental unit...
	citeseer.ist.psu.edu /396485.html (558 words)

	Robert Boltje
		Also, the presence of a Mackey functor structure on the ideal class groups of number fields in a fixed Galois extension provides relations between these class groups.
		The ideal class group is an invariant which measures how close the ring of integers in a number field is to having unique factorization into primes.
		Presently, Robert Boltje is interested in the conjectures of Alperin, Dade, and Broué in the representation theory of finite groups.
	www.math.ucsc.edu /Faculty/Boltje.html (203 words)

	Structure Operations
		The function also returns a map between the group and the magma of quadratic forms of the associated discriminant.
		The structure of the class group of the order O or the maximal order of the quadratic field K, as a sequence of integers giving the abelian invariants.
		The unit group of the order O or the maximal order of the quadratic field K, as an abelian group, together with a map to the order (or field).
	www.math.ufl.edu /help/magma/text377.html (561 words)

	Cyclotomic Integers of Prescribed Absolute Value and the Class Group (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: We obtain a new method for the study of class groups of cyclotomic fields by investigating cyclotomic integers of prescribed absolute value.
		Explicit subgroups of the classgroup C modulo the class group C + of the maximal real subfield are exhibited and lower bounds on their orders are derived.
		1 Abhyankar's lemma and the class group (context) - Cornell - 1979
	citeseer.ist.psu.edu /schmidt97cyclotomic.html (466 words)

	[No title]
		The Euler system of cyclotomic units (and a more general Kolyvagin system counterpart) is a collection of "derivatives" of cyclotomic units in a cyclotomic number field whose factorizations gives bounds on the size of the ideal class group.
		A major result from this line of attack is that the size of the p-part of the chi-component of the ideal class group equal the size of the p-part of the chi-component of the group of units mod cyclotomic units.
		Loosely speaking, Lubin's conjecture states that if we have the case where an invertible power series commutes with a non-invertible one, there must exist a formal group over that extension such that both the aforementioned invertible power series and non-invertible power series are both endomorphisms of the same formal group.
	math.arizona.edu /~mcleman/Research/Research.html (655 words)

	Ideal class group (Site not responding. Last check: 2007-11-07)
		In general the bound isn't sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.
		All is still licensed under the GNU FDL.
		For a moment, the surprised tories stood mute in alarm.html">alarm and doubt, from the latter to the door and windows, as if weighing the chances turned, and drawing.html">drawing and flourishing their knives behind them, sprang leg, and hurling him back with stunning effect upon the floor.
	www.termsdefined.net /id/ideal-class-group.html (1458 words)

	Ideal Class Groups
		Using these relations, a generating set for the ideal class group is derived (via matrix echelonization), and in the final step it is checked that the correct orders for the generators are found.
		A point worth noting is that the verification routine checks the result under the assumption that the given bound was large enough; hence if an insufficiently large bound was given the result may still be incorrect despite passing the verification stage.
		Verify the correctness of the class group as found by the function ClassGroup.
	www.math.wisc.edu /help/magma/text449.html (965 words)

	Galois Groups and Greenberg's Conjecture (Site not responding. Last check: 2007-11-07)
		In this thesis we consider the structure of a certain infinite Galois group over K, the field of p-th roots of unity.
		The main result of this dissertation shows this Iwasawa module to be torsion free for a large class of cyclotomic fields.
		As one consequence, we provide a large class of examples of cyclotomic fields which do not admit free pro-p-extensions of maximal possible rank r2+1.
	www.math.utexas.edu /~marshall/dis.html (129 words)

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