# Divisor Theory

Birkhäuser Boston, Jan 1, 1990 - Mathematics - 184 pages
Kronecker's theory of divisors, like Dedekind's theory of ideals, is a broad generalization of Kummer's original theory of 'ideal prime factors' in cyclotomic fields. Kronecker never gave a fully elaborated treatment of the theory of divisors, and the few writers who have done so (Heinrich Weber, Hermann Weyl) have failed to show its simplicity, its scope, and its advantages over the now-familliar Dedekian theory. This book gives a full development of a general theory of divisors (including some topics not treated by Kronecker, such as ramification and the different), together with applications to algebraic number theory and to the theory of algebraic curves. An appendix on differentials makes possible the statement and proof of the Riemann-Roch theorem for curves. Contents Preface Part 0. A Theorem of Polynomial Algebra Part 1. The General Theory -1.1 Introduction. -1.2 Natural Rings. -1.3 On Existence. -1.4 Preliminaries. -1.5 The Basic Theory -1.6 Definitions. -1.7 What is a Divisor? -1.8 -1.9 -1.10 First Propositions. -1.11 The Main Proposition -1.12 Concluding Corollaries. -1.13 Testing for Divisibility -1.14 The Group of Nonzero Divisors -1.15 Greatest Common Divisors -1.16 Ambient Fields. -1.17 Norms. -1.18 Factorization of Divisors. -1.19 -1.20 Two Basic Theorems -1.21 The Divisor Class Group. -1.22 Prime Factorization and Normal Extensions. -1.23 -1.24 Rings of Values -1.25 Primitive Elements, Norms, and Traces -1.26 -1.27 -1.28 Differents. -1.29 -1.30 Discriminants. -1.31 -1.32 Ramification. Part 2. Applications to Algebraic Number Theory -2.1 Factorization into Primes. -2.2 -2.3 -2.4 A Factorization Method -2.5 Examples -2.6 Integral Bases. -2.7 Proof of the Theorem -2.8 Dedekind's Descriminant Theorem -2.9 Differents and Discriminants -2.10 -2.11 Cyclotomic Fields. -2.12 Quadratic Reciprocity Part 3. Applications to the Theory of Algebraic Curves -3.1 Function Fields. -3.2 -3.3 Parameters and Constants. -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -3.10 Global Divisors. -3.11 Numerical Extensions. -3.12 -3.13 -3.14 The Idea of a Place -3.15 -3.16 Local Parameters at a Place -3.17 Relative norms -3.18 -3.19 A Divisor is a Product of Powers of Places. -3.20 Presentation of Places. -3.21 Degree of a Divisor. -3.22 A Characteristic of Places. -3.23 Dimension of a Divisor. -3.24 The Genus of a Function Field. -3.25 Abel's Theorem. -3.26 The Genus as a Limit. -3.27 A Converse of Abel's Theorem. -3.28 The Divisor Class Group. -3.29 Examples Appendix -A.1 Introduction. -A.2 Definitions and First Propositions. -A.3 Orders and Residues of Differentials. -A.4 The Sum of the Residues is Zero. -A.5 Holomorphic Differentials. -A.6 Integral Bases. -A.7 Normal Bases. -A.8 -A.9 The Dual of a Normal Basis. -A.10 Construction of Holomorphic Differentials. -A.11 Examples. -A.12 The Riemann-Roch Theorem for Integral Divisors. -A.13 Riemann-Roch for Reciprocals of Integral Divisors. -A.14 General Case of the Riemann-Roch Theorem. References

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### About the author (1990)

Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990) and Linear Algebra (1995). Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new. In 1980 he was awarded the Steele Prize for mathematical exposition for the Riemann and Fermat books.