Preface Preface to the previous edition 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 ofcurves 1.3 Athirdviewpoint：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 Fieldinclusions and rationalmaps 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 Discretevaluation rings 3.4 Simultaneous approximations with several discrete valuation rings 3.5 Extensions of discrete valuation rings 3.6 Completions ofdiscrete valuation rings 3.6.1 On valuation rings and geometric points again 3.6.2 Normsonvector 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 Proofofthe double implication『weak form'? strongform A.3 Twofurther 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 Dedekinddomains C.2 FactOrization C.3 Further notes to appendices Index References
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