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Chapman and Hall/CRC
632 pages | 17 B/W Illus.
Many basic ideas of algebra and number theory intertwine, making it ideal to explore both at the same time. Certain Number-Theoretic Episodes in Algebra focuses on some important aspects of interconnections between number theory and commutative algebra. Using a pedagogical approach, the author presents the conceptual foundations of commutative algebra arising from number theory. Self-contained, the book examines situations where explicit algebraic analogues of theorems of number theory are available.
Coverage is divided into four parts, beginning with elements of number theory and algebra such as theorems of Euler, Fermat, and Lagrange, Euclidean domains, and finite groups. In the second part, the book details ordered fields, fields with valuation, and other algebraic structures. This is followed by a review of fundamentals of algebraic number theory in the third part. The final part explores links with ring theory, finite dimensional algebras, and the Goldbach problem.
“…starts by covering the classical theorems, then describes the integral domain of rational integers. Euclidean domains, rings of polynomials and former power series, … He analyzes the relevance of algebraic structures to number theory in such topics as ordered fields, fields with valuation and other algebraic structures, …”
— In Scitech Book News, December 2006
“This book is focused on some important aspects of interrelations between number theory and commutative algebra. The book is divides into four parts; each chapter starts with a historical overview and closes with illustrative examples. … the book can be recommended to anyone interested in these domains. …”
— In EMS Newsletter, March 2007
ELEMENTS OF NUMBER THEORY AND ALGEBRA
Theorems of Euler, Fermat and Lagrange
Historical perspective
Introduction
The quotient ring Z / rZ
An elementary counting principle
Fermat’s two squares theorem
Lagrange’s four squares theorem
Diophantine equations
Notes with illustrative examples
Worked-out examples
The Integral Domain of Rational Integers
Historical perspective
Introduction
Ordered integral domains
Ideals in a commutative ring
Irreducibles and primes
GCD domains
Notes with illustrative examples
Worked-out examples
Euclidean Domains
Historical perspective
Introduction
Z as a Euclidean domain
Quadratic number fields
Almost Euclidean domains
Notes with illustrative examples
Worked-out examples
Rings of Polynomials and Formal Power Series
Historical perspective
Introduction
Polynomial rings
Elementary arithmetic functions
Polynomials in several indeterminates
Ring of formal power series
Finite fields and irreducible polynomials
More about irreducible polynomials
Notes with illustrative examples
Worked-out examples
The Chinese Remainder Theorem and the Evaluation of Number of Solutions of a Linear Congruence with Side Conditions
Historical perspective
Introduction
The Chinese Remainder theorem
Direct products and direct sums
Even functions (mod r)
Linear congruences with side conditions
The Rademacher formula
Notes with illustrative examples
Worked-out examples
Reciprocity Laws
Historical perspective
Introduction
Preliminaries
Gauss lemma
Finite fields and quadratic reciprocity law
Cubic residues (mod p)
Group characters and the cubic reciprocity law
Notes with illustrative examples
A comment by W. C. Waterhouse
Worked-out examples
Finite Groups
Historical perspective
Introduction
Conjugate classes of elements in a group
Counting certain special representations of a group element
Number of cyclic subgroups of a finite group
A criterion for the uniqueness of a cyclic group of order r
Notes with illustrative examples
A worked-out example
An example from quadratic residues
THE RELEVANCE OF ALGEBRAIC STRUCTURES TO
NUMBER THEORY
Ordered Fields, Fields with Valuation and Other Algebraic Structures
Historical perspective
Introduction
Ordered fields
Valuation rings
Fields with valuation
Normed division domains
Modular lattices and Jordan-Hölder theorem
Non-commutative rings
Boolean algebras
Notes with illustrative examples
Worked-out examples
The Role of the Möbius Function—Abstract Möbius Inversion
Historical perspective
Introduction
Abstract Möbius inversion
Incidence algebra of n × n matrices
Vector spaces over a finite field
Notes with illustrative examples
Worked-out examples
The Role of Generating Functions
Historical perspective
Introduction
Euler’s theorems on partitions of an integer
Elliptic functions
Stirling numbers and Bernoulli numbers
Binomial posets and generating functions
Dirichlet series
Notes with illustrative examples
Worked-out examples
Catalan numbers
Semigroups and Certain Convolution Algebras
Historical perspective
Introduction
Semigroups
Semicharacters
Finite dimensional convolution algebras
Abstract arithmetical functions
Convolutions in general
A functional-theoretic algebra
Notes with illustrative examples
Worked-out examples
A GLIMPSE OF ALGEBRAIC NUMBER THEORY
Noetherian and Dedekind Domains
Historical perspective
Introduction
Noetherian rings
More about ideals
Jacobson radical
The Lasker-Noether decomposition theorem
Dedekind domains
The Chinese remainder theorem revisited
Integral domains having finite norm property
Notes with illustrative examples
Worked-out examples
Algebraic Number Fields
Historical perspective
Introduction
The ideal class group
Cyclotomic fields
Half-factorial domains
The Pell equation
The Cakravala method
Dirichlet’s unit theorem
Notes with illustrative examples
Formally real fields
Worked-out examples
SOME MORE INTERCONNECTIONS
Rings of Arithmetic Functions
Historical perspective
Introduction
Cauchy composition (mod r)
The algebra of even functions (mod r)
Carlitz conjecture
More about zero divisors
Certain norm-preserving transformations
Notes with illustrative examples
Worked-out examples
Analogues of the Goldbach Problem
Historical perspective
Introduction
The Riemann hypothesis
A finite analogue of the Goldbach problem
The Goldbach problem in Mn(Z)
An analogue of Goldbach theorem via polynomials over finite fields
Notes with illustrative examples
A variant of Goldbach conjecture
An Epilogue: More Interconnections
Introduction
On commutative rings
Commutative rings without maximal ideals
Infinitude of primes in a PID
On the group of units of a commutative ring
Quadratic reciprocity in a finite group
Worked-out examples
True/False Statements: Answer Key
Index of Some Selected Structure Theorems/Results
Index of Symbols and Notations
Bibliography
Subject Index
Index of names
Each chapter includes exercises and references.
