Lecture Notes in Mathematics

# The Decomposition of Primes in Torsion Point Fields

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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.

• Introduction

Pages 1-4

$29.95 • Decomposition Laws Pages 5-24$29.95
• Elliptic Curves

Pages 25-39

$29.95 • Elliptic Modular Curves Pages 41-58$29.95
• Torsion Point Fields

Pages 59-86

$29.95 ### Buy this book eBook$34.99
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• Digitally watermarked, DRM-free
• Included format: PDF
• ebooks can be used on all reading devices
Softcover \$49.95
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• ISBN 978-3-540-42035-4
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## Bibliographic Information

Bibliographic Information
Book Title
The Decomposition of Primes in Torsion Point Fields
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
1761
2001
Publisher
Springer-Verlag Berlin Heidelberg
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-540-44949-2
DOI
10.1007/b80624
Softcover ISBN
978-3-540-42035-4
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 148
Topics