Certain Number-Theoretic Episodes In Algebra
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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The integral domain of rational integers
Rings of polynomials and formal power series
The Chinese Remainder Theorem and the evaluation of number of solutions of a linear congruence with side conditions
THE RELEVANCE OF ALGEBRAIC STRUCTURES TO NUMBER THEORY
A GLIMPSE OF ALGEBRAIC NUMBER THEORY
Noetherian and Dedekind domains
Algebraic number fields
SOME MORE INTERCONNECTIONS
Rings of arithmetic functions
Analogues of the Goldbach problem
An epilogue More interconnections