Local fields
by Cassels, J. W. S., (John William Scott)
The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-containe…
Read More ↓Title Statements
- Full title :Local fields / J.W.S. Cassels.
Summary
The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-contained introduction to local fields. After a general introduction, attention centres on the p-adic numbers and their use in number theory. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a p-adic variable. This book will appeal to undergraduates, and even amateurs, interested in number theory, as well as to graduate students.
Read More ↓Edition Statements
- Publication Date:1986.
Publication Statements
- Publisher(s):Cambridge University Press
- Place of Publication:Cambridge [Cambridgeshire]
- Language:eng
- Publication Date:1986.
The Physical Object
- Description:1 online resource (xiv, 360 pages) : : illustrations
Edition Identifiers
- ISBN:9781139171885, 1139171887, 9781107087644, 1107087643, 9781107093850, 1107093856
- oclc:ocn817968639
- LC Classification:QA247
- Dewey Decimal:512/.3
Table of Contents
Cover; Series Page; Title; Copyright; PREFACE; CONTENTS; LEITFADEN; NOTATIONAL CONVENTIONS; CHAPTER ONE: INTRODUCTION; 1 VALUATIONS; 2 REMARKS; 3 AN APPLICATION; CHAPTER TWO: GENERAL PROPERTIES; 1 DEFINITIONS AND BASICS; 2 VALUATIONS ON THE RATIONALS; 3 INDEPENDENCE OF VALUATIONS; 4 COMPLETENESS; 5 FORMAL SERIES AND A THEOREM OF EISENSTEIN; CHAPTER THREE: ARCHIMEDEAN VALUATIONS; 1 INTRODUCTION; 2 SOME LEMMAS; 3 COMPLETION OF PROOF; CHAPTER FOUR: NON-ARCHIMEDEAN VALUATIONS. SIMPLE PROPERTIES; 1 DEFINITIONS AND BASICS; 2 AN APPLICATION TO FINITE GROUPS OF RATIONAL MATRICES; 3 HENSEL'S LEMMA
4 ELEMENTARY ANALYSIS5 A USEFUL EXPANSION; 6 AN APPLICATION TO RECURRENT SEQUENCES; CHAPTER FIVE: EMBEDDING THEOREM; 1 INTRODUCTION; 2 THREE LEMMAS; 3 PROOF OF THEOREM; 4 APPLICATION. A THEOREM OF SELBERG; 5 APPLICATION. THE THEOREM OF MAHLER AND LECH; CHAPTER SIX: TRANSCENDENTAL EXTENSIONS. FACTORIZATION; 1 INTRODUCTION; 2 GAUSS' LEMMA AND EISENSTEIN IRREDUCIBILITY; 3 NEWTON POLYGON; 4 FACTORIZATION OF PURE POLYNOMIALS; 5 WEIERSTRASS PREPARATION THEOREM; CHAPTER SEVEN: ALGEBRAIC EXTENSIONS (COMPLETE FIELDS); 1 INTRODUCTION; 2 UNIQUENESS; 3 EXISTENCE; 4 RESIDUE CLASS FIELDS; 5 RAMIFICATION
6 DI…
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