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Topic: Class field theory

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	PlanetMath: algebraic number theory
		As an introduction, the reader should be comfortable with the basic theory of rational and irrational numbers, and its complementary entry, the basic theory of algebraic and transcendental numbers.
		Class field theory studies the abelian extensions of number fields.
		Class field theory and the Artin map can be presented in terms of idèles and adèles.
	planetmath.org /encyclopedia/AlgebraicNumberTheory.html (0 words)

	[No title]
		Class field for H intersect H' is composite of class fields for H and H'; uniqueness of class fields; every class field is contained in a ray class field.
		Summary of infinite Galois theory: the Krull topology on Aut L; the Galois group of the fixed field of S is the closed subgroup generated by S; the fundamental theorem.
		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 (0 words)

	PlanetMath: ray class field
		Thus we obtain a characterization of the ray class field of conductor
		Cross-references: Hilbert class field, root of unity, primitive, characterization, degree, residue fields, extension, prime, proposition, properties, ideal, Artin map, ramify, prime ideals, divisible, integral ideal, ring of integers, number fields, abelian extension, finite
		This is version 2 of ray class field, born on 2003-08-24, modified 2003-08-26.
	planetmath.org /encyclopedia/RayClassField.html (0 words)

	Introduction
		Class field theory is concerned with the classification of all abelian extensions of a given field.
		Class field theory classifies all abelian extensions of a given number field k in terms of quotients of ray class groups.
		Class field theory asserts the existence of some module m and a subgroup H such that G isomorphic to Cl_m/H by the Artin-map.
	www.umich.edu /~gpcc/scs/magma/text660.htm (0 words)

	Search Results for theory
		Chevalley also published Theory of Distributions (1951), Introduction to the theory of algebraic functions of one variable (1951), The algebraic theory of spinors (1954), Class field theory (1954), The construction and study of certain important algebras (1955), Fundamental concepts of algebra (1956) and Foundations of algebraic geometry (1958).
		Stable homotopy theory and generalized homology (1974) comprises of three lecture courses, one on the algebra of stable operations in complex cobordism delivered in 1967, the second on complex cobordism theory delivered in 1970, and the third on stable homotopy and generalized homology theories delivered in 1971.
		This theory, called the caloric theory, was based on two axioms, namely that the heat in the universe is conserved and that the heat in a substance is a function of the state of the substance.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=theory&CONTEXT=1 (18793 words)

	Springer Online Reference Works
		The basic theorems in class field theory were formulated and proved in particular cases by L.
		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).
		For example, class field theory is used in proving the existence of infinite class field towers (see Tower of fields).
	eom.springer.de /c/c022370.htm (0 words)

	Open Questions: Algebraic Number Theory
		Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa.
		Class field theory, which we are almost ready to discuss, deals directly with all these issues, and provides answers to many of the questions which arise.
		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.
	www.openquestions.com /oq-ma018.htm (0 words)

	3.2.4 Number Theory
		The number theory group at Oklahoma State University has established a thriving program of research, including a regular seminar series featuring lectures of both a research and expository nature by the resident number theorists, as well as frequent lectures by distinguished young and senior number theorists from around the country.
		Number theory is famed not just for the beauty of its theorems, but for the enormous wealth and variety of techniques involved in discovering and proving these theorems.
		Our faculty is prepared to offer courses in algebraic number theory, class field theory, analytic number theory, the arithmetic of elliptic curves as well as other arithmetic algebraic varieties, p-adic analysis, automorphic and modular forms, discrete subgroups of algebraic groups, computational number theory, as well as many other subfields of number theory.
	www.math.okstate.edu /grad/brief-hbk/3_2_4Number_Theory.html (395 words)

	[No title]
		I participated also at the seminar of Number Theory lead by Professor N. Popescu devoted to algebraic number theory, analytic number theory, class field theory and theory of algebraic functions.
		Fields of interest: Number theory with special interest in analytic number theory, algebraic numbers and functions, transcendental numbers, local class field theory, diofantine equations.
		A finiteness theorem for a class of exponential congruences, Proc.
	www.imar.ro /~mvajaitu (942 words)

	Primes of the Form x² + ny²: Fermat, Class Field Theory, and Complex ...
		In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of class field theory and its intimate connection with complex multiplication.
		Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication.
		A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication.
	www.ecampus.com /bk_detail.asp?referrer=CJ&isbn=0471190799 (383 words)

	The Math Forum - Math Library - Fields
		Preprints on the arithmetic of fields, Galois theory, model theory of fields, and related topics: text abstracts and papers in DVI format may be downloaded.
		Field theory looks at sets, such as the real number line, on which all the usual arithmetic properties hold, including, now, those of division.
		The study of multiple fields through Galois theory is important for the study of polynomial equations, and thus has applications to number theory and group theory.
	mathforum.org /library/topics/fields (0 words)

	[No title]
		The aim of the class-field theory is to describe all Abelian extensions of a given field k and at its source lies the Kronecker--Weber Theorem, which solves this problem for k=Q, the field of rational numbers (L. Kronecker [,], H.
		This led later to the theory of Dedekind domains and culminated in the theory of ideals in commutative rings.
		Modern class-field theory begins with the invention of ideles by C. Chevalley [] who in C. Chevalley [] reinterpreted classical class-field theory in terms of ideles, using the theory of associative algebras.
	www1.elsevier.com /homepage/saj/523281/h10.htm (0 words)

	Class Field Theory
		The class field theory of local fields classifies abelian extensions of local field in a way similar to the way global class field theory deals with extensions of number fields and global function fields.
		Since the multiplicative group of a local field is far better understood than the ideal group of a global field, the theory is much more explicit and easier in the local case.
		The principal units of a p-adic ring or field R are elements of the form 1 + pi Z_R where pi is a uniformizing element of R and Z_R is the ring of integers.
	magma.maths.usyd.edu.au /magma/htmlhelp/text778.htm (0 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.
		The ray class group modulo D is a quotient of the group of divisors that are coprime to D modulo certain principal divisors.
		Since the ray class field modulo m is always an infinite field extension containing the algebraic closure of the constant field, this returns Infinity.
	magma.maths.usyd.edu.au /magma/htmlhelp/text733.htm (0 words)

	Number Theory Seminar Title and Abstract - University of Sydney
		Class Field Theory and the Kronecker Weber Theorem
		The Main Theorem of Complex Multiplication for abelian varieties is a generalisation of the Kronecker-Weber theorem which states that every abelian extension of $\Q$ is contained in a cyclotomic field, and also the result of Weber and Takagi which says that every abelian extension of an imaginary quadratic field is generated by singular moduli, i.e.
		This week I will lay the groundwork from class field theory necessary to state and prove the main theorem seen in Shimura's book.
	www.maths.usyd.edu.au /u/NumTheorySeminar/abstracts/2006.06.01_ward.html (0 words)

	Algebra and Number Theory Faculty (Site not responding. Last check: )
		Efraim Armendariz (efraim@math.utexas.edu): Research interests include the general structure theory of noncommucative rings and their modules, with an emphasis on rings satisfying a polynomial identity and von Neumann regular rings; radical properties and torsion theories for ring and module categories.
		His future research plans are to continue developing the theory of asymptotic and essential prime divisors and their applications, and to study projective equivalence.
		David Saltman (saltman@math.utexas.edu): Research interests include Brauer group theory and division algebras, with an emphasis on invariant theory of groups acting on fields, rationality of invariant fields, the center of the generic division algebra, and division algebras over p-adic curves and their geometry.
	www.ma.utexas.edu /dev/math/Research/Field_of_Interest/Algebra_and_Number_Theory.html (393 words)

	Math 254 (Number Theory)
		Math 254B took a detailed look at class field theory, the theory of abelian extensions of number fields, which extends the reciprocity laws of Gauss, Legendre, Hilbert et al.
		The Berkeley Number Theory Seminar meets Wednesdays from 3:10 to 4:00 PM in 891 Evans, and sometimes on Friday at the same time and place.
		The Number Theory Web is the home of all things number-theoretic on the Web.
	www-math.mit.edu /~kedlaya/math254b.html (400 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 class field defined by m has a Galois groups that is isomorphic to R/ker(m^(-1)) under the Artin map.
	www.math.niu.edu /help/math/magmahelp/text672.html (763 words)

	Springer Online Reference Works
		The structure of the group of units of a field was elucidated by P.
		The class number can be explicitly described in terms of other field constants (the regulator, the discriminant and the degree of the field).
		For a discussion of the relations between the ideal-theoretic formulations and the modern idèle based class field theory [a2] is recommended.
	eom.springer.de /A/a011600.htm (0 words)

	Referativni Zhurnal Classification
		Groupoids, etc. 271.17.17.33.17 Special classes of groupoids 271.17.17.33.21 Groupoids with complemented structures 271.17.17.33.31 Quasigroups 271.17.17.33.31.17 Isotopies and homotopies of quasigroups 271.17.17.33.31.21 Identities and generalized identities on quasigroups 271.17.17.33.31.31 Loops 271.17.19 Rings and modules 271.17.19.15 Methods of mathematical logic in rings and modules 271.17.19.19 Associative rings and algebras 271.17.19.19.15 Structure of rings 271.17.19.19.15.17 Ideals in rings.
		Matrix theory 271.17.29.19.17 Determinants and their generalizations 271.17.29.19.21 Matrix equations 271.17.29.19.25 Eigenvalues of matrices 271.17.29.19.33 Special classes of matrices 271.17.29.21 Systems of linear equations and inequalities 271.17.29.31 Polylinear algebra.
		Theory of finite differences 271.23.19.15.17 Finite-difference equations 271.23.19.15.17.21 Recurrent relations and series 271.23.19.19 Functional equations and inequalities 271.23.21 Integral transformations, operational calculus 271.23.21.17 Laplace transform 271.23.21.19 Fourier integral and Fourier transform 271.23.21.21 Other integral transformations and their inversions.
	www.ams.org /mathweb/Classif/RZhClassification.html (0 words)

	A Brief Guide to Algebraic Number Theory - Cambridge University Press
		This is an account of Algebraic Number Theory, a field which has grown to touch many other areas of pure mathematics.
		It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites.
		Class field theory; Appendix; Exercises; Suggested further reading.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521004233 (0 words)

	[No title] (Site not responding. Last check: )
		From: N.P.Strickland@sheffield.ac.uk Date: Wed, 10 May 2000 12:53:50 +0100 Subject: postdoc position Research Associateship in ------------------------- Equivariant Elliptic Cohomology and Class Field Theory ====================================================== A two year position is available at the University of Sheffield for a research associate to work on a project in equivariant elliptic cohomology and class field theory.
		I would be interested in candidates with a strong background in either (a) computational number theory with emphasis on elliptic curves and class field theory or (b) elliptic cohomology, stable homotopy theory and/or generalised cohomology of classifying spaces of finite groups.
		Neil Strickland --------------------------------------------------------------------- The appointment is for two years and involves working with Dr Neil Strickland on the EPSRC funded project ``Equivariant elliptic cohomology and class field theory'' The idea is to use class field theory to calculate some rings related to the equivariant elliptic cohomology of finite groups.
	www.lehigh.edu /dmd1/public/www-data/ns510 (501 words)

	Math 776 Syllabus
		Math 776, as the continuation of Math 676, 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.
		Then we cover the formulations of the statements of global class field theory, for number fields, and local class field theory, for p-adic fields.
	www.math.lsa.umich.edu /~lagarias/m776-0.html (0 words)

	[No title]
		That's "simple" if you know class field theory: the class field H of the quadratic field k has Galois group isomorphic to Z/N, and the base extension k/Q is cyclic and acts on Z/N via the Artin isomorphism.
		What this means is that the nontrivial isomorphism S of k/Q acts on a generator of Z/N in the same way as it acts on the class c generating the class group.
		But c^S = c^{-1}, since c^{S+1} = 1 (the norm of an ideal is principal since the base field Q has class number 1), and voila: the complete extension H/Q has the dihedral group.
	www.math.niu.edu /~rusin/known-math/99/dihedral_gal (335 words)

	Math 249
		Over two quarters, this course will focus on the class field theory, including the construction of the Weil group and the theories of Hecke and Artin L-functions.
		Hilbert in his 1897 Zahlbericht gave a profound analysis of Kummer's work leading to the notion of a class field.
		Noether determined the structure of division algebras over local and global fields, allowing new foundations for class field theory.
	sporadic.stanford.edu /bump/math249.html (0 words)

	Algebraic Number Theory
		to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry.
		The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook....
		Algebraic Integers.- The Theory of Valuations.- Riemann-Roch Theory.- Abstract Class Field Theory.- Local Class Field Theory.- Global Class Field Theory.- Zeta Functions and L-series.- Bibliography.- Index.
	www.booksmatter.com /b3540653996.htm (196 words)

	Front: [math.NT/0702379] Quantum statistical mechanics and class field Theory
		To appear in the proceedings of the summer school "Geometric and topological methods for quantum field theory", Villa de Leyva 2005.
		Abstract: We survey some results relating noncommutative geometry to the class field theory of number fields.
		These results appear within the context of quantum statistical mechanics where some arithmetic properties of a given number field can be realized in terms of the structure of equilibrium states of a quantum statistical mechanical system.
	front.math.ucdavis.edu /math.NT/0702379 (0 words)

	GT Monographs: Volume 3
		This monograph is the result of the conference on higher local fields held in Muenster, August 29 to September 5, 1999.
		The aim is to provide an introduction to higher local fields (more generally complete discrete valuation fields with arbitrary residue field) and render the main ideas of this theory (Part I), as well as to discuss several applications and connections to other areas (Part II).
		Given an arithmetic scheme, there is a higher local field associated to a flag of subschemes on it.
	www.msp.warwick.ac.uk /gt/gtmcontents3.html (447 words)

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