目录
 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
 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