Number Theory by Benjamin Fine, Gerhard Rosenberger - 9783319438733

Birkhauser Verlag AG

Discover the fascinating world of number theory with "Number Theory: An Introduction via the Density of Primes." This comprehensive textbook, now in its second edition, offers a unique approach by focusing on the density and properties of prime numbers. It provides a solid foundation in standard number theory concepts, including the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. This edition expands on key areas with new coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key features include: * A thorough introduction to analytic number theory with full proofs of Dirichlet's Theorem and the Prime Number Theorem. * Concise treatment of algebraic number theory, detailing primes, prime factorizations in algebraic number fields, and unique factorization of ideals. * Discussion of the AKS algorithm, demonstrating that primality testing is a polynomial-time problem. * Exploration of interesting ancillary topics such as primality testing, cryptography, Fermat and Mersenne numbers, and Carmichael numbers. Designed for upper-level undergraduates and beginning graduate students, this book requires only a basic knowledge of calculus, multivariable calculus, and linear algebra, with necessary concepts from abstract algebra and complex analysis introduced as needed. Its user-friendly style, historical context, and range of exercises make it ideal for both self-study and classroom use. Product Summary: Hardcover book on Number Theory by Benjamin Fine and Gerhard Rosenberger, published by Birkhäuser Cham. Features 413 pages, first edition, English language version, and includes 11 b/w and 1 color illustration. ISBN: 9783319438733.