This is an elementary introduction to algebra and number theory. Starting with a recalling of the preliminaries of groups, rings and fields, the topics in algebra portion consists of polynomial rings, UFD, PID and Euclidean domains, field extensions, modules and Dedckind domains. In the number theory part, apart from covering elementary congruence results, the laws of quadratic reciprocity and basics of algebraic number fields, this book attempts to give glimpse of various aspects of the subject. Thus, Warning's and Chevally's theorems in the section of finite fields and many results of additive number theory are strewn all over. Some of the additive number theoretic results like derivation of Lagrange's four square theorem from Minkowski's result in geometry of numbers, are given. By adding some remarks and comments, which are beyond the scope of this book, and with the references in the bibliography, attempts are been made to encourage the students to learn beyond this material.
Sukumar Das Adhikari.: The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, India
Preface/Notations and terminologies/Preliminaries: Groups, rings and Fields/The integers/Polynomial Rings/Exercise Set A/Rings and fields revisited/Factorization/Gauss lemma and Eisenstein criterion/Field Extension/Quadratic reciprocity law/Exercise Set B/Modules/Gaussian integers and the ring Z[ -5]/Algebraic number fields-I/Exercise Set C/Dedekind Domains/Algebraic number fields-II/Exercise Set D/Quadratic Fields/Solutions to selected exercises/Appendix: Lucas-Lehmer test/Bibliography/Index