# The Decomposition of Primes in Torsion Point Fields [electronic resource] / edited by Clemens Adelmann.

- Publication:
- Berlin, Heidelberg : Springer Berlin Heidelberg, 2001.
- Series:
- Lecture Notes in Mathematics, 0075-8434 ; 1761

Lecture Notes in Mathematics, 0075-8434 ; 1761 - Format/Description:
- Book

1 online resource (VIII, 148 pages) - Subjects:
- Number theory.

Geometry, algebraic. - Local subjects:
- Number Theory. (search)

Algebraic Geometry. (search) - System Details:
- text file PDF
- Summary:
- 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.
- Contents:
- Introduction

Decomposition laws

Elliptic curves

Elliptic modular curves

Torsion point fields

Invariants and resolvent polynomials

Appendix: Invariants of elliptic modular curves; L-series coefficients a p; Fully decomposed prime numbers; Resolvent polynomials; Free resolution of the invariant algebra. - Contributor:
- Adelmann, Clemens. editor., Editor,

SpringerLink (Online service) - Contained In:
- Springer eBooks
- Other format:
- Printed edition:

Printed edition: - ISBN:
- 9783540449492
- Publisher Number:
- 10.1007/b80624 doi
- Access Restriction:
- Restricted for use by site license.
- Online:
- Connect to full text

http://hdl.library.upenn.edu/1017.12/2329628 -
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