| The Decomposition of Primes in Torsion Point Fields 2001 Edition Contributor(s): Adelmann, Clemens (Author) |
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ISBN: 3540420355 ISBN-13: 9783540420354 Publisher: Springer OUR PRICE: $47.45 Product Type: Paperback - Other Formats Published: May 2001 |
| Additional Information |
| BISAC Categories: - Mathematics | Algebra - General - Medical - Mathematics | Number Theory |
| Dewey: 512.4 |
| LCCN: 2001031199 |
| Series: Lecture Notes in Computer Science, |
| Physical Information: 0.34" H x 6.14" W x 9.21" (0.51 lbs) 148 pages |
| Descriptions, Reviews, Etc. |
| Publisher 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 |