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Full Description
Spanning elementary, algebraic, and analytic approaches, this book provides an introductory overview of essential themes in number theory. Designed for mathematics students, it progresses from undergraduate-accessible material requiring only basic abstract algebra to graduate-level topics demanding familiarity with algebra and complex analysis. The first part covers classical themes: congruences, quadratic reciprocity, partitions, cryptographic applications, and continued fractions with connections to quadratic Diophantine equations. The second part introduces key algebraic tools, including Noetherian and Dedekind rings, then develops the finiteness of class groups in number fields and the analytic class number formula. It also examines quadratic fields and binary quadratic forms, presenting reduction theory for both definite and indefinite cases. The final section focuses on analytic methods: L-series, primes in arithmetic progressions, and the Riemann zeta function. It addresses the Prime Number Theorem and explicit formulas of von Mangoldt and Riemann, equipping students with foundational knowledge across number theory's major branches.
Contents
Preface; Part I. Elementary Methods: 1. Congruences and primes; 2. Continued fractions; 3. Euclidean and principal ideal domains; Part II. Algebraic Methods: 4. Some commutative algebra; 5. Integrality; 6. Ideal class groups and units; 7. Quadratic fields and binary quadratic forms; 8. Cyclotomic fields; Part III. Analytic Methods: 9. Dirichlet series; 10. The Riemann zeta function; 11. The prime number theorem; Bibliography; Index.
