Arithmetics. Transl. from the French by Sarah Carr (Q2275441)

This book is the faithful translation into English of the author's text ``Arithmétique'' [Paris: Calvage et Mounet (2008; Zbl 1135.11001)], the French original of which has been reviewed very thoroughly right after its publication in 2008. Apart from a number of minor corrections and a couple of examples added in Chapter 3, the book has been left totally unchanged. Therefore, as for the contents, the structure, and the appreciation of this very special textbook in the field of general number theory and its applications, we may utterly refer to our review of the original French edition (Zbl 1135.11001), and that in every respect. However, it might be quite appropriate to recall here some of the particular features of this outstanding textbook, which obviously also prompted its quick translation into English. Geared toward graduate students at the master's level (M1 and M2), the book provides a thorough and lively introduction to various fundamental aspects of both classical and contemporary arithmetical theories, together with some of their most important applications and current research developments. In contrast to most textbooks in number theory, the present treatise emphasizes a broad panoramic approach to the subject, thereby combining an elementary introduction to some classical topics, in the first part, with a more advanced outlook to various themes of current mathematical research in the second half of the book. More precisely, the first four chapters are devoted to the following introductory topics: 1. Finite structures (\((\mathbb Z/n\mathbb Z)^*\) and \(\mathbb F^*_q\); Jacobi and Legendre symbols; Gauss sums; applications to the number of solutions of equations). 2. Applications (arithmetic algorithms; cryptography and RSA; primality tests; factorization of integers; error-correcting codes). 3. Algebra and Diophantine equations (sums and squares; Fermat's equation for the exponents 3 and 4; Pell's equation; rings of algebraic integers; geometry of numbers and applications in algebraic number theory). 4. Analytic number theory (prime numbers and estimates; summary of holomorphic functions; Dirichlet series and the zeta function; characters and Dirichlet's theorem on primes in arithmetic progressions; the prime number theorem and its analytic proof). The more advanced and topical themes are discussed in the subsequent two chapters covering the following material: 5. Elliptic curves (group law on a cubic; heights; the Mordell-Weil theorem; Siegel's theorem; elliptic curves over \(\mathbb C\); elliptic curves over a finite field; the \(L\)-function on an elliptic curve). 6. Developments and open problems (the number of solutions of equations over finite fields; Diophantine equations and algebraic geometry; \(p\)-adic numbers; transcendental numbers and Diophantine approximation; the \((a,b,c)\)-conjecture; some remarkable Dirichlet series and Wiles's theorem). Three appendices on recent factorization algorithms in computational number theory, elementary projective geometry, and Galois theory of number fields, respectively, complete the material of the main text. Altogether, the book under review is both an excellent introduction and a truly irresistible invitation to number theory in its various fascinating aspects. Covering a vast spectrum of central topics in the field, with hundreds of carefully selected examples and exercises woven into the main text, this book is a masterpiece of expository writing in mathematics. Its current translation into English will certainly augment both the worldwide popularity and usefulness of this remarkable textbook.