Goldschmidt, David M. Algebraic functions and projective curves. (English) Zbl 1034.14011 Graduate Texts in Mathematics 215. New York, NY: Springer (ISBN 0-387-95432-5/hbk). xvi, 179 p. (2003). Index: Chapter 1. Background (Valuations; completions; differential forms; residues; exercises). – Chapter 2. Function fields (Divisors and adeles; Weil differentials; elliptic functions; geometric function fields; residue and duality; exercises). – Chapter 3. Finite extensions (Norm and conorm; scalar extensions; the different; singular prime divisors; Galois extensions; hyperelliptic functions; exercises). – Chapter 4. Projective curves (Projective varieties; maps to \(\mathbb {P}^n\); projective embeddings; Weierstrass points; plane curves; exercises). – Chapter 5. Zeta functions (The Euler product; the functional equation; the Riemann hypothesis; exercises). – Appendix: Elementary field theoryThis is a very nice algebraic introduction to the theory of algebraic curves (no geometry) with full, clear and simple proofs. It should be very useful for workers in coding theory. Reviewer: Edoardo Ballico (Povo) Cited in 22 Documents MSC: 14H05 Algebraic functions and function fields in algebraic geometry 11R42 Zeta functions and \(L\)-functions of number fields 11R58 Arithmetic theory of algebraic function fields 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G50 Applications to coding theory and cryptography of arithmetic geometry Keywords:function fields; Weil differentials; Weierstrass points; Riemann hypothesis; zeta functions; coding theory PDFBibTeX XMLCite \textit{D. M. Goldschmidt}, Algebraic functions and projective curves. New York, NY: Springer (2003; Zbl 1034.14011) Full Text: DOI