Algebra and Number Theory: A Selection of Highlights

Description

This two-volume set collects and presents some fundamentals of mathematics in an entertaining and performing manner. The present volume examines many of the most important basic results in algebra and number theory, along with their proofs, and also their history. Contents The natural, integral and rational numbers Division and factorization in the integers Modular arithmetic Exceptional numbers Pythagorean triples and sums of squares Polynomials and unique factorization Field extensions and splitting fields Permutations and symmetric polynomials Real numbers The complex numbers, the Fundamental Theorem of Algebra and polynomial equations Quadratic number fields and Pell’s equation Transcendental numbers and the numbers e and π Compass and straightedge constructions and the classical problems Euclidean vector spaces An exciting collection of fundamental results in algebra and number theory Aimed at lecturers, teachers, and students of mathematics, and at all mathematically interested Contains many exercises related to the content of each chapter Preface Contents 1. The natural, integral and rational numbers 2. Division and factorization in the integers 3. Modular arithmetic 4. Exceptional numbers 5. Pythagorean triples and sums of squares 6. Polynomials and unique factorization 7. Field extensions and splitting fields 8. Permutations and symmetric polynomials 9. Real numbers 10. The complex numbers, the Fundamental Theorem of Algebra and polynomial equations 11. Quadratic number fields and Pell’s equation 12. Transcendental numbers and the numbers e and π 13. Compass and straightedge constructions and the classical problems 14. Euclidean vector spaces Bibliography Index