Number Theory in Function Fields

Michael Rosen (Autor)

Buch | Softcover

358 Seiten

2010 | Softcover reprint of hardcover 1st ed. 2002
Springer-Verlag New York Inc.
978-1-4419-2954-9 (ISBN)

Artikel merken

Elementary number theory is concerned with the arithmetic properties of the ring of integers, Z, and its field of fractions, the rational numbers, Q. Early on in the development of the subject it was noticed that Z has many properties in common with A = IF[T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating this by presenting, for example, analogues of the little theorems of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in an arithmetic progression. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con sidering finite algebraic extensions K of Q, which are called algebraic num ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K.

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.

Erscheint lt. Verlag	3.12.2010
Reihe/Serie	Graduate Texts in Mathematics ; 210
Zusatzinfo	XI, 358 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	1-4419-2954-1 / 1441929541
ISBN-13	978-1-4419-2954-9 / 9781441929549
Zustand	Neuware