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Focusing on the interconnections between number theory and commutative algebra, this self-contained book presents the conceptual foundations of commutative algebra arising from number theory. Using a pedagogical approach, the author examines situations where explicit algebraic analogues of some theorems of number theory are available. Beginning with elements of number theory and algebra, the book studies ordered fields, fields with valuation inversion formulae, generating functions, and algebraic number theory. It explores links with ring theory and the polynomial analogue of the Goldbach problem. Coverage includes the Chinese remainder theorem, reciprocity laws, finite groups, abstract Mobius inversion, rings of arithmetic functions, and finite dimensional algebras.
List of contents
Elements of Number Theory and Algebra. The Relevance of Algebraic Structures to Number Theory. A Glimpse of Algebraic Number Theory. Some More Interconnections.
About the author
R, Sivaramakrishnan; Sivaramakrishnan, R
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.
