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  • © 2001

The Decomposition of Primes in Torsion Point Fields

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1761)

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Table of contents (6 chapters)

  1. Front Matter

    Pages I-V
  2. Introduction

    Pages 1-4
  3. Decomposition Laws

    Pages 5-24
  4. Elliptic Curves

    Pages 25-39
  5. Torsion Point Fields

    Pages 59-86
  6. Back Matter

    Pages 107-140

About this book

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.

Keywords

  • algebra
  • algebraic number theory
  • elliptic curve
  • invariant theory
  • modular form
  • number theory
  • prime number
  • torsion

Editors and Affiliations

  • Institute of Applied Mathematics Dept. of Applied Algebra, Technical University Braunschweig, Braunschweig, Germany

    Clemens Adelmann

Bibliographic Information

  • Book Title: The Decomposition of Primes in Torsion Point Fields

  • Editors: Clemens Adelmann

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/b80624

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2001

  • Softcover ISBN: 978-3-540-42035-4Published: 22 May 2001

  • eBook ISBN: 978-3-540-44949-2Published: 11 October 2004

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 148

  • Topics: Number Theory, Algebraic Geometry

Buying options

eBook USD 34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions