本书系统呈现了赋值论的基础内容，深入探讨其在代数几何与数论领域的重要应用。全书分为两个核心部分： 第一部分聚焦于代数几何基础理论，重点阐释代数曲线函数域相关概念。这部分内容不仅构建起代数几何的基础框架，更为后续赋值论的深入探讨奠定坚实基础。通过循序渐进的讲解方式，帮助读者逐步理解代数几何中的核心概念与理论体系，为后续研究做好充分准备。 第二部分围绕赋值环及其完备性展开系统研究。该部分在沿用经典研究方法的基础上，创新性地融入前沿学术成果。作者巧妙地将传统理论与最新研究进展相结合，既保证了内容的系统性和严谨性，又满足了专业读者对前沿知识的探索需求。无论是初学者还是资深研究者，都能从这部分内容中收获启发，感受赋值论研究的独特魅力与理论深度。

前辅文
Chapter 1 Introduction: Generalities on algebraic functions of one variable
  1.1 A first viewpoint for algebraic functions: Riemann surfaces
  1.2 A second viewpoint: geometry of curves
  1.3 A third viewpoint: fields
Chapter 2 Basic notions on function fields of one variable
  2.1 Function fields of one variable, rational fields
   2.1.1 Unirational and rational function fields
  2.2 Function fields (of one variable) define (plane) curves
  2.3 Algebraic varieties, rings of regular functions
  2.4 Field inclusions and rational maps
   2.4.1 Birational equivalence
Chapter 3 Valuation rings
  3.1 Valuation rings and places
   3.1.1 Some significant examples
  3.2 Existence and extensions of valuation rings
   3.2.1 Some applications of Theorem 3.1
  3.3 Discrete valuation rings
  3.4 Simultaneous approximations with several discrete valuation rings
  3.5 Extensions of discrete valuation rings
  3.6 Completions of discrete valuation rings
   3.6.1 On valuation rings and geometric points again
   3.6.2 Norms on vector spaces
  3.7 Notes to Chapter 3
Appendix A Hilbert’s Nullstellensatz
  A.1 Generalities
   A.1.1 Review of preliminaries on algebraic sets
  A.2 Proofs
   A.2.1 Proof of the double implication ‘weak form’ ? ‘strong form’
  A.3 Two further applications
Appendix B Puiseux series
  B.1 Field of definition, convergence, and Eisenstein theorem
  B.2 Algebraic functions and differential equations
Appendix C Discrete valuation rings and Dedekind domains
  C.1 Dedekind domains
  C.2 Factorization
  C.3 Further notes to appendices
Index
References