Book contents
- Frontmatter
- Preface
- Contents
- Leitfaden
- Notational conventions
- Chapter 1 Introduction
- Chapter 2 General properties
- Chapter 3 Archimedean valuations
- Chapter 4 Non archimedean valuations. Simple properties
- Chapter 5 Embedding theorem
- Chapter 6 Transcendental extensions. Factorization
- Chapter 7 Algebraic extensions (complete fields)
- Chapter 8 p-adic fields
- Chapter 9 Algebraic extensions (incomplete fields)
- Chapter 10 Algebraic number fields
- Chapter 11 Diophantine equations
- Chapter 12 Advanced analysis
- Chapter 13 A theorem of Borel and Dwork
- Appendix A Resultants and discriminants
- Appendix B Norms, traces and characteristic polynomials
- Appendix C Minkowski's convex body theorem
- Appendix D Solution of equations in finite fields
- Appendix E Zeta and L-functions at negative integers
- Appendix F Calculation of exponentials
- References
- Index
Chapter 8 - p-adic fields
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Preface
- Contents
- Leitfaden
- Notational conventions
- Chapter 1 Introduction
- Chapter 2 General properties
- Chapter 3 Archimedean valuations
- Chapter 4 Non archimedean valuations. Simple properties
- Chapter 5 Embedding theorem
- Chapter 6 Transcendental extensions. Factorization
- Chapter 7 Algebraic extensions (complete fields)
- Chapter 8 p-adic fields
- Chapter 9 Algebraic extensions (incomplete fields)
- Chapter 10 Algebraic number fields
- Chapter 11 Diophantine equations
- Chapter 12 Advanced analysis
- Chapter 13 A theorem of Borel and Dwork
- Appendix A Resultants and discriminants
- Appendix B Norms, traces and characteristic polynomials
- Appendix C Minkowski's convex body theorem
- Appendix D Solution of equations in finite fields
- Appendix E Zeta and L-functions at negative integers
- Appendix F Calculation of exponentials
- References
- Index
Summary
INTRODUCTION
In this Chapter we study the complete valued fields which will arise when we look at algebraic number fields in Chapter 10. The first two sections are straightforward applications and amplifications of the results of the previous Chapter and prepare the way for Chapter 10. The remainder of the Chapter gives a couple of results which are important in the further development of the theory but which are not required further in this book.
We start by defining the fields we shall be considering:
DEFINITION 1.1. Let the field k be complete with respect to the (nonarch.) valuation | |. We say that k is a p-adic field if
(i) k has characteristic 0
(ii) | | is discrete
(iii) the residue class field ρ is finite.
We can give at once an alternative characterization:
LEMMA 1.1. The valued field k is a p-adic field if and only if it is a finite extension of ℚpfor some p.
Proof. (i) Suppose that k is a finite extension of ℚp. Then it is a p-adic field by Lemma 4.1 and 5.1 of Chapter 7.
(ii) Let k be a p-adic field. Then k ⊃ ℚ by (i) of the definition. Since the residue class field is finite by (iii) of the definition, it has characteristic p for some prime p. Hence the valuation on k induces a valuation equivalent to the p-adic valuation.
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- Local Fields , pp. 144 - 164Publisher: Cambridge University PressPrint publication year: 1986