Certain Number-Theoretic Episodes In Algebra

1st Edition

R Sivaramakrishnan, Sivaramakrishnan R

Chapman and Hall/CRC
Published September 22, 2006
Reference - 632 Pages - 17 B/W Illustrations
ISBN 9780824758950 - CAT# DK3054

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Features

  • Highlights the analogues of the fundamental theorem of arithmetic (FTA)
  • Studies the ring Z of integers that is the Euclidean domain having “double-remainder property” (drp)
  • Proves the Chinese remainder theorem, reciprocity laws, and quadratic reciprocity in a finite group
  • Examines algebraic structures, inversion formulae, the role of generating functions, and convolution algebras
  • Discusses Noetherian and Dedekind domains that pertain to elements of algebraic number theory
  • Provides a survey of rings of arithmetic functions, Carlitz conjecture, and finite dimensional algebras
  • Gives instances of “infinitude of primes” in ring theory
  • Explains the polynomial analogue of the Goldbach problem
  • Summary

    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.

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