Examines algebra and number theory topics at a level appropriate for a one-semester, graduate-level course
Assumes a basic knowledge of set theory
Establishes various number theoretic results throughout the sections on algebra
Contains a proof of the Lucas-Lehmer test
Includes a separate section on quadratic reciprocity that includes a complete proof of the theorem
Stimulates further study through the author's remarks and a thorough bibliography
An Introduction to Commutative Algebra and Number Theory is an elementary introduction to these subjects. Beginning with a concise review of groups, rings and fields, the author presents topics in algebra from a distinctly number-theoretic perspective and sprinkles number theory results throughout his presentation. The topics in algebra include polynomial rings, UFD, PID, and Euclidean domains; and field extensions, modules, and Dedekind domains.
In the section on number theory, in addition to covering elementary congruence results, the laws of quadratic reciprocity and basics of algebraic number fields, this book gives glimpses into some deeper aspects of the subject. These include Warning's and Chevally's theorems in the finite field sections, and many results of additive number theory, such as the derivation of LaGrange's four-square theorem from Minkowski's result in the geometry of numbers.
With addition of remarks and comments and with references in the bibliography, the author stimulates readers to explore the subject beyond the scope of this book.
Table of Contents
PRELIMINARIES: GROUPS, RINGS, AND FIELDS Groups Rings and Fields THE INTEGERS Divisibility The Fundamental Theorem of Arithmetic The Chinese Remainder Theorem POLYNOMIAL RINGS Rings of Polynomials Division in Polynomial Rings Exercise Set A RINGS AND FIELDS REVISITED Characteristic of a Ring Wilson's Theorem A Result on Vector Spaces FACTORIZATION Divisibility UFD and PID Euclidean Domains GAUSS LEMMA AND EISENSTEIN CRITERION Gauss Lemma Eisenstein Criterion FIELD EXTENSIONS Algebraic Extensions Normal Extensions Separable Extensions Finite Fields QUADRATIC RECIPROCITY LAW Exercise Set B MODULES Basic Definitions A Result on Finitely Generated Modules Noetherian Modules Modules over a PID Some Special Results GAUSSIAN INTEGERS AND THE RING Z [v-5] Gaussian Integers The Ring Z [v-5] ALGEBRAIC NUMBER FIELDS-I Integral Dependence Integers in Number Fields Exercise Set C DEDEKIND DOMAINS Fractional Ideals Properties of Dedekind Domains ALGEBRAIC NUMBER FIELDS-II Class Groups Discriminants Some Results in Geometry of Numbers An Estimation Dirichlet's Unit Theorem Exercise Set D QUADRATIC FIELDS Integral Bases and Discriminants Splitting of Rational Primes The Group of Units Norm-Euclidean Number Fields Solutions to Selected Exercises Appendix: Lucas-Lehmer Test Bibliography Index
CPD consists of any educational activity which helps to maintain and develop knowledge, problem-solving, and technical skills with the aim to provide better health care through higher standards. It could be through conference attendance, group discussion or directed reading to name just a few examples.
Use certain CRC Press medical books to get your CPD points up for revalidation. We provide a free online form to document your learning and a certificate for your records.