Certain Number-Theoretic Episodes In Algebra

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.