Certain number-theoretic episodes in algebra /

Certain number-theoretic episodes in algebra /

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Bibliographic Details
Author / Creator:Sivaramakrishnan, R., 1936-
Imprint:Boca Raton, FL : Chapman & Hall/CRC, c2007.
Description:632 p. ; 25 cm.
Language:English
Series:Pure and applied mathematics ; 286
Monographs and textbooks in pure and applied mathematics ; 286.
Subject:
Algebraic number theory.
Number theory.
Algebraic number theory.
Number theory.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/6118987
Hidden Bibliographic Details
ISBN:0824758951 (alk. paper)
Notes:Includes bibliographical references (p. [615]-619) and indexes.
Standard no.:9780824758950
Table of Contents:
  • Part I. Elements of Number Theory and Algebra
  • 1. Theorems of Euler, Fermat and Lagrange
  • Historical perspective
  • 1.1. Introduction
  • 1.2. The quotient ring Z/rZ
  • 1.3. An elementary counting principle
  • 1.4. Fermat's two squares theorem
  • 1.5. Lagrange's four squares theorem
  • 1.6. Diophantine equations
  • 1.7. Notes with illustrative examples
  • 1.8. Worked-out examples
  • Exercises
  • References
  • 2. The integral domain of rational integers
  • Historical perspective
  • 2.1. Introduction
  • 2.2. Ordered integral domains
  • 2.3. Ideals in a commutative ring
  • 2.4. Irreducibles and primes
  • 2.5. GCD domains
  • 2.6. Notes with illustrative examples
  • 2.7. Worked-out examples
  • Exercises
  • References
  • 3. Euclidean domains
  • Historical perspective
  • 3.1. Introduction
  • 3.2. Z as a Euclidean domain
  • 3.3. Quadratic number fields
  • 3.4. Almost Euclidean domains
  • 3.5. Notes with illustrative examples
  • 3.6. Worked-out examples
  • Exercises
  • References
  • 4. Rings of polynomials and formal power series
  • Historical perspective
  • 4.1. Introduction
  • 4.2. Polynomial rings
  • 4.3. Elementary arithmetic functions
  • 4.4. Polynomials in several indeterminates
  • 4.5. Ring of formal power series
  • 4.6. Finite fields and irreducible polynomials
  • 4.7. More about irreducible polynomials
  • 4.8. Notes with illustrative examples
  • 4.9. Worked-out examples
  • Exercises
  • References
  • 5. The Chinese Remainder Theorem and the evaluation of number of solutions of a linear congruence with side conditions
  • Historical perspective
  • 5.1. Introduction
  • 5.2. The Chinese Remainder Theorem
  • 5.3. Direct products and direct sums
  • 5.4. Even functions (mod r)
  • 5.5. Linear congruences with side conditions
  • 5.6. The Rademacher formula
  • 5.7. Notes with illustrative examples
  • 5.8. Worked-out examples
  • Exercises
  • References
  • 6. Reciprocity laws
  • Historical perspective
  • 6.1. Introduction
  • 6.2. Preliminaries
  • 6.3. Gauss lemma
  • 6.4. Finite fields and quadratic reciprocity law
  • 6.5. Cubic residues (mod p)
  • 6.6. Group characters and the cubic reciprocity law
  • 6.7. Notes with illustrative examples
  • 6.8. A comment
  • 6.9. Worked-out examples
  • Exercises
  • References
  • 7. Finite groups
  • Historical perspective
  • 7.1. Introduction
  • 7.2. Conjugate classes of elements in a group
  • 7.3. Counting certain special representations of a group element
  • 7.4. Number of cyclic subgroups of a finite group
  • 7.5. A criterion for the uniqueness of a cyclic group of order r
  • 7.6. Notes with illustrative examples
  • 7.7. A worked-out example
  • 7.8. An example from quadratic residues
  • Exercises
  • References
  • Part II. The Relevance of Algebraic Structures to Number Theory
  • 8. Ordered fields, fields with valuation and other algebraic structures
  • Historical perspective
  • 8.1. Introduction
  • 8.2. Ordered fields
  • 8.3. Valuation rings
  • 8.4. Fields with valuation
  • 8.5. Normed division domains
  • 8.6. Modular lattices and Jordan-Holder theorem
  • 8.7. Non-commutative rings
  • 8.8. Boolean algebras
  • 8.9. Notes with illustrative examples
  • 8.10. Worked-out examples
  • Exercises
  • References
  • 9. The role of the Mobius function- Abstract Mobius inversion
  • Historical perspective
  • 9.1. Introduction
  • 9.2. Abstract Mobius inversion
  • 9.3. Incidence algebra of n x n matrices
  • 9.4. Vector spaces over a finite field
  • 9.5. Notes with illustrative examples
  • 9.6. Worked-out examples
  • Exercises
  • References
  • 10. The role of generating functions
  • Historical perspective
  • 10.1. Introduction
  • 10.2. Euler's theorems on partitions of an integer
  • 10.3. Elliptic functions
  • 10.4. Stirling numbers and Bernoulli numbers
  • 10.5. Binomial posets and generating functions
  • 10.6. Dirichlet series
  • 10.7. Notes with illustrative examples
  • 10.8. Worked-out examples
  • 10.9. Catalan numbers
  • Exercises
  • References
  • 11. Semigroups and certain convolution algebras
  • Historical perspective
  • 11.1. Introduction
  • 11.2. Semigroups
  • 11.3. Semicharacters
  • 11.4. Finite dimensional convolution algebras
  • 11.5. Abstract arithmetical functions
  • 11.6. Convolutions in general
  • 11.7. A functional-theoretic algebra
  • 11.8. Notes with illustrative examples
  • 11.9. Worked-out examples
  • Exercises
  • References
  • Part III. A Glimpse of Algebraic Number Theory
  • 12. Noetherian and Dedekind domains
  • Historical perspective
  • 12.1. Introduction
  • 12.2. Noetherian rings
  • 12.3. More about ideals
  • 12.4. Jacobson radical
  • 12.5. The Lasker-Noether decomposition theorem
  • 12.6. Dedekind domains
  • 12.7. The Chinese remainder theorem revisited
  • 12.8. Integral domains having finite norm property
  • 12.9. Notes with illustrative examples
  • 12.10. Worked-out examples
  • Exercises
  • References
  • 13. Algebraic number fields
  • Historical perspective
  • 13.1. Introduction
  • 13.2. The ideal class group
  • 13.3. Cyclotomic fields
  • 13.4. Half-factorial domains
  • 13.5. The Pell equation
  • 13.6. The Cakravala method
  • 13.7. Dirichlet's unit theorem
  • 13.8. Notes with illustrative examples
  • 13.9. Formally real fields
  • 13.10. Worked-out examples
  • Exercises
  • References
  • Part IV. Some More Interconnections
  • 14. Rings of arithmetic functions
  • Historical perspective
  • 14.1. Introduction
  • 14.2. Cauchy composition (mod r)
  • 14.3. The algebra of even functions (mod r)
  • 14.4. Carlitz conjecture
  • 14.5. More about zero divisors
  • 14.6. Certain norm-preserving transformations
  • 14.7. Notes with illustrative examples
  • 14.8. Worked-out examples
  • Exercises
  • References
  • 15. Analogues of the Goldbach problem
  • Historical perspective
  • 15.1. Introduction
  • 15.2. The Riemann hypothesis
  • 15.3. A finite analogue of the Goldbach problem
  • 15.4. The Goldbach problem in M[subscript n](Z)
  • 15.5. An analogue of Goldbach theorem via polynomials over finite fields
  • 15.6. Notes with illustrative examples
  • 15.7. A variant of Goldbach conjecture
  • Exercises
  • References
  • 16. An epilogue: More interconnections
  • Introduction
  • 16.1. On commutative rings
  • 16.2. Commutative rings without maximal ideals
  • 16.3. Infinitude of primes in a PID
  • 16.4. On the group of units of a commutative ring
  • 16.5. Quadratic reciprocity in a finite group
  • 16.6. Worked-out examples
  • References
  • True/False statements: Answer key
  • Index of some selected structure theorems/results
  • Index of symbols and notations
  • Bibliography
  • Subject index
  • Index of names
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