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 9 - Algebraic extensions (incomplete 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
Let K/k be a finite algebraic extension and let | | be a valuation on k. We do not suppose that k is complete and ask ourselves what extensions, if any, there are of | | to K. We shall encounter this problem also when | | is archimedean. For most of this chapter we can consider arch, and non-arch, valuations together. The situation we shall be considering is usually evoked by the keyword semi-local.
Suppose that the valuation | |1 on K extends | | and let K1 be the completion of K with respect to it. Then K1 contains the completion k of k with respect to | |. A basis {Bi} of K/k clearly generates K1 as a k-vector space. There is, however, no reason to expect that the Bi, considered as elements of K1, will be linearly independent over k, and we can conclude only that [K1 : k] ≤ [K : k]. Multiplication gives K1 a natural structure as K-module.
We shall also require the tensor product k ⊗k, K. This can be described concretely, if non-canonically, as follows. Let B1,…,Bn be a basis for K/k. There are cijℓ ∈ k such that
Then k ⊗k, K is an n-dimensional k-vector space with a basis which we identify with the B1:
It has a ring structure, multiplication being defined by (1.1) and by k-linearity. We identify K in k ⊗k, K with the linear combinations of the Bi with coefficients in k.
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- Local Fields , pp. 165 - 188Publisher: Cambridge University PressPrint publication year: 1986