Certain Number-Theoretic Episodes In Algebra

Couverture
CRC Press, 22 sept. 2006 - 632 pages
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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Table des matières

Theorems of Euler Fermat and Lagrange
3
The integral domain of rational integers
29
Euclidean domains
47
Rings of polynomials and formal power series
73
The Chinese Remainder Theorem and the evaluation of number of solutions of a linear congruence with side conditions
105
Reciprocity laws
139
Finite groups
177
THE RELEVANCE OF ALGEBRAIC STRUCTURES TO NUMBER THEORY
203
A GLIMPSE OF ALGEBRAIC NUMBER THEORY
373
Noetherian and Dedekind domains
375
Algebraic number fields
435
SOME MORE INTERCONNECTIONS
481
Rings of arithmetic functions
483
Analogues of the Goldbach problem
525
An epilogue More interconnections
577
BIBLIOGRAPHY
615

Ordered fields fields with valuation and other algebraic structures
205
The role of the Möbius function Abstract Möbius inversion
261
The role of generating functions
291
Semigroups and certain convolution algebras
339

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Page vii - Acknowledgement The author wishes to express his deep sense of gratitude to Dr.
Page vi - Department of Science and Technology, Ministry of Science and Technology, Government of India, New Delhi, is gratefully acknowledged.

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