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Cover: 9780387953359 | Number Theory in Function Fields | Michael Rosen | Buch | xi | 2002
Bild: 9780387953359 | Number Theory in Function Fields | Michael Rosen | Buch | xi | 2002
Cover: 9780387953359 | Number Theory in Function Fields | Michael Rosen | Buch | xi | 2002
Bild: 9780387953359 | Number Theory in Function Fields | Michael Rosen | Buch | xi | 2002
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Number Theory in Function Fields
Buch von Michael Rosen
Sprache: Englisch

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Beschreibung
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.
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.
Zusammenfassung
This book studies the relationship between number theory in algebraic number fields and algebraic function fields. Because function fields are a bit different from number fields, even the experienced number theorist will learn from this book. Algebraic geometers will like the book, since the geometry of curves over an algebraically closed field is both pretty and elementary. Michael Rosen is the author of the successful book "A Classical Introduction to Modern Number Theory." He is the recipient of the 1999 Chauvenet Prize for his article "Niels Hendrik Abel and Equations of the Fifth Degree."
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2002
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Graduate Texts in Mathematics
Inhalt: xi
358 S.
ISBN-13: 9780387953359
ISBN-10: 0387953353
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Rosen, Michael
Hersteller: Springer New York
Springer US, New York, N.Y.
Graduate Texts in Mathematics
Maße: 241 x 160 x 26 mm
Von/Mit: Michael Rosen
Erscheinungsdatum: 08.01.2002
Gewicht: 0,735 kg
Artikel-ID: 104625568
Zusammenfassung
This book studies the relationship between number theory in algebraic number fields and algebraic function fields. Because function fields are a bit different from number fields, even the experienced number theorist will learn from this book. Algebraic geometers will like the book, since the geometry of curves over an algebraically closed field is both pretty and elementary. Michael Rosen is the author of the successful book "A Classical Introduction to Modern Number Theory." He is the recipient of the 1999 Chauvenet Prize for his article "Niels Hendrik Abel and Equations of the Fifth Degree."
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2002
Fachbereich: Geometrie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Graduate Texts in Mathematics
Inhalt: xi
358 S.
ISBN-13: 9780387953359
ISBN-10: 0387953353
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Rosen, Michael
Hersteller: Springer New York
Springer US, New York, N.Y.
Graduate Texts in Mathematics
Maße: 241 x 160 x 26 mm
Von/Mit: Michael Rosen
Erscheinungsdatum: 08.01.2002
Gewicht: 0,735 kg
Artikel-ID: 104625568
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