Number Theory in Function Fields / by Michael Rosen
(Graduate Texts in Mathematics ; 210)
| データ種別 | 電子ブック |
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| 出版者 | New York, NY : Springer New York : Imprint: Springer |
| 出版年 | 2002 |
| 本文言語 | 英語 |
| 大きさ | XI, 358 p : online resource |
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| 内容注記 | 1 Polynomials over Finite Fields 2 Primes, Arithmetic Functions, and the Zeta Function 3 The Reciprocity Law 4 Dirichlet L-Series and Primes in an Arithmetic Progression 5 Algebraic Function Fields and Global Function Fields 6 Weil Differentials and the Canonical Class 7 Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem 8 Constant Field Extensions 9 Galois Extensions — Hecke and Artin L-Series 10 Artin’s Primitive Root Conjecture 11 The Behavior of the Class Group in Constant Field Extensions 12 Cyclotomic Function Fields 13 Drinfeld Modules: An Introduction 14 S-Units, S-Class Group, and the Corresponding L-Functions 15 The Brumer-Stark Conjecture 16 The Class Number Formulas in Quadratic and Cyclotomic Function Fields 17 Average Value Theorems in Function Fields Appendix: A Proof of the Function Field Riemann Hypothesis Author Index |
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| 一般注記 | Elementary number theory is concerned with arithmetic properties of the ring of integers. Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting, for example, analogues of the theorems of Fermat and Euler, Wilsons theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlets theorem on primes in an arithmetic progression. After presenting the required foundational material on function fields, the later chapters explore the analogy between global function fields and algebraic number fields. A variety of topics are presented, including: the ABC-conjecture, Artins conjecture on primitive roots, the Brumer-Stark conjecture, Drinfeld modules, class number formulae, and average value theorems. The first few chapters of this book are accessible to advanced undergraduates. The later chapters are designed for graduate students and professionals in mathematics and related fields who want to learn more about the very fruitful relationship between number theory in algebraic number fields and algebraic function fields. In this book many paths are set forth for future learning and exploration. Michael Rosen is Professor of Mathematics at Brown University, where hes been since 1962. He has published over 40 research papers and he is the co-author of A Classical Introduction to Modern Number Theory, with Kenneth Ireland. He received the Chauvenet Prize of the Mathematical Association of America in 1999 and the Philip J. Bray Teaching Award in 2001 |
| 著者標目 | *Rosen, Michael author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Algebraic geometry LCSH:Algebra LCSH:Field theory (Physics) LCSH:Number theory FREE:Mathematics FREE:Number Theory FREE:Algebraic Geometry FREE:Field Theory and Polynomials |
| 分 類 | DC23:512.7 |
| 巻冊次 | ISBN:9781475760460 
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| ISBN | 9781475760460 |
| URL | http://dx.doi.org/10.1007/978-1-4757-6046-0 |
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