Elementary Theory

Teachers have been using Elementary Music Theory for over 25 years as a beginning theory method. This edition is filled with exercises for elementary level students. It compliments any beginner method and is great to use right from the very first lesson. Helpful diagrams and images accompany each lesson and combine with a variety of engaging learning activities to reinforce essential skills and knowledge.

Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised. The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.

The theory of p-adic and classic modular forms, and the study of arithmetic and p-adic L-functions has proved to be a fruitful area of mathematics over the last decade. Professor Hida has given courses on these topics in the USA, Japan, and in France, and in this book provides the reader with an elementary but detailed insight into the theory of L-functions. The presentation is self contained and concise, and the subject is approached using only basic tools from complex analysis and cohomology theory. Graduate students wishing to know more about L-functions will find that this book offers a unique introduction to this fascinating branch of mathematics.

Basic treatment includes existence theorem for solutions of differential systems where data is analytic, holomorphic functions, Cauchy's integral, Taylor and Laurent expansions, more. Exercises. 1973 edition.

Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.

An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and more.