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Lectures on Number Theory

The book is intended as a manual for a half-year introductory course of number theory. The aim is to present the theory to the extent which is indispensable to every mathematics student, regardless of their future specialization. The target points include the local methods, the Quadratic Reciprocity Law, the Prime Number Theorem, unique factorization of ideals in rings of algebraic integers in finite extensions of~Q, units in the rings of integers in quadratic extensions of~Q, and decomposition of primes in rings of integers. The authors' main aim was to show that the beauty of number theory lies in the fact that on the one hand it enriches other domains of mathematics, and on the other hand it makes frequent use of their methods and results. Each chapter is complemented with problems, of different complexity. Some solutions and hints are given at the end of the book.

Introduction  7

1. Basic algebraic properties of integers 9
2. Congruences mod m and basic algebraic properties of the rings Zm 16
3. Local method  23
4. Gaussian integers 30
5. Representation of prime numbers as x2 + dy2 34
6. Legendre symbol and the Quadratic Reciprocity Law . 38
7. Applications to cryptography 51
8. Classical problems of number theory . 55
9. Prime Number Theorem and its proof 64
10. Integral elements. Trace and norm . 70
11. The rings of integers in finite extensions of Q  76
12. Decomposition of ideals in Dedekind domains  82
13. The rings of integers in quadratic and cyclotomic extensions of Q 88
14. Splittings of ideals. A new look at QRL 100
15. Hints and solutions to selected problems 108

Bibliography 119
Index  120

122 pages, Paperback