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Abelian extension
About: Abelian extension is a research topic. Over the lifetime, 1953 publications have been published within this topic receiving 25508 citations.
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01 Jan 1967
TL;DR: In this article, the authors present an algebraic extension of the Field of Quotients of an Integral Domain (FQDN) to the field of rings and fields.
Abstract: A Few Preliminaries. Mathematics and Proofs.Sets and Relations.Mathematical Induction.Complex and Matrix Algebra. 1. Groups and Subgroups. Binary Operations.Finite-State Machines (Automata).Isomorphic Binary Structures.Groups.Subgroups.Cyclic Groups and Generators.Cayley Digraphs. 2. More Groups and Cosets. Groups of Permutations.Automata.Orbits, Cycles, and the Alternating Groups.Plane Isometries.Cosets and the Theorem of Lagrange.Direct Products and Finitely Generated Abelian Groups.Periodic Functions, Plane Isometries.Binary Linear Codes. 3. Homomorphisms and Factor Groups. Homomorphisms.Factor Groups.Factor-group Computations and Simple Groups.Series of Groups.Group Action on a Set.Applications of G-sets to Counting. 4. Advanced Group Theory. Isomorphism Theorems Proof of the Jordan-Holder Theorem.Sylow Theorem.Applications of the Sylow Theory.Free Abelian Groups.Free Groups.Group Presentations. 5. Introduction to Rings and Fields. Rings and Fields.Integral Domains.Fermat's and Euler's Theorems.The Field of Quotients of an Integral Domain.Rings of Polynomials.Factorization of Polynomials over a Field.Noncommutative Examples.Ordered Rings and Fields. 6. Factor Rings and Ideals. Homomorphisms and Factor Rings.Prime and Maximal Ideals.Grobner Bases for Ideals. 7. Factorization*. Unique Factorization Domains.Euclidean Domains.Gaussian Integers and Norms. 8. Extension Fields. Introduction to Extension Fields.Vector Spaces.Algebraic Extensions.Geometric Constructions.Finite Fields.Additional Algebraic Structures. 9. Automorphisms and Galois Theory. Automorphisms of Fields.The Isomorphism Extension Theorem.Splitting Fields.Separable Extensions.Totally Inseparable Extensions.Galois Theory.Illustrations of Galois Theory.Cyclotomic Extensions.Insolvability of the Quintic. Bibliography. Notations. Answers To Odd-Numbered Exercises Not Requiring Proofs. Index. *Not required for the remainder of the text.
624 citations
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1969, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
589 citations
TL;DR: In this article, the realization of certain types of Chevalley groups as the Galois group of extensions of cyclotomic fields is studied, and a criterion for algebraic curve to be defined over an algebraic number field is given.
Abstract: This paper is devoted to the realization of certain types of Chevalley groups as the Galois group of extensions of certain cyclotomic fields. In addition, a criterion for an algebraic curve to be defined over an algebraic number field is given. Bibliography: 11 titles.
551 citations
484 citations