Certain Number-Theoretic Episodes In Algebra: 1st Edition (Hardback) book cover

Certain Number-Theoretic Episodes In Algebra

1st Edition

By R Sivaramakrishnan, Sivaramakrishnan R

Chapman and Hall/CRC

632 pages | 17 B/W Illus.

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Description

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.

Reviews

“…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

Table of Contents

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory