The decomposition of primes in torsion point fields
Author(s)
Bibliographic Information
The decomposition of primes in torsion point fields
(Lecture notes in mathematics, 1761)
Springer-Verlag, c2001
Available at / 76 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||176178800286
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
DC21:519/AD332070534713
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Note
Includes bibliographical references (p. [135]-136) and index
Description and Table of Contents
Description
It is an historical goal of algebraic number theory to relate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively described in terms of the base ?eld. Suitable structures are the prime ideals of the ring of integers of the considered number ?eld. By examining the behaviouroftheprimeidealswhenembeddedintheextension?eld,su?cient information should be collected to distinguish the given extension from all other possible extension ?elds. The ring of integers O of an algebraic number ?eld k is a Dedekind ring. k Any non-zero ideal in O possesses therefore a decomposition into a product k of prime ideals in O which is unique up to permutations of the factors. This k decomposition generalizes the prime factor decomposition of numbers in Z Z. In order to keep the uniqueness of the factors, view has to be changed from elements of O to ideals of O . k k Given an extension K/k of algebraic number ?elds and a prime ideal p of O , the decomposition law of K/k describes the product decomposition of k the ideal generated by p in O and names its characteristic quantities, i. e. K the number of di?erent prime ideal factors, their respective inertial degrees, and their respective rami?cation indices. Whenlookingatdecompositionlaws,weshouldinitiallyrestrictourselves to Galois extensions. This special case already o?ers quite a few di?culties.
Table of Contents
Decomposition Laws.- Elliptic Curves.- Elliptic Modular Curves.- Torsion Point Fields.- Invariants and Resolvent Polynomials.
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