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**An introduction to number theory.**
*(English)*
Zbl 1089.11001

Graduate Texts in Mathematics 232. London: Springer (ISBN 1-85233-917-9/hbk). x, 294 p. (2005).

The book under review contains several topics which are usually not brought together in an introductory text. The book is meant to give a broad introduction to advanced undergraduate students with no knowledge of number theory. Therefore, the authors hit many subjects but do not go too deep into details. Each chapter contains many exercises and historical notes. The authors require a modest background in complex analysis and elementary algebra.

In my opinion, because so many topics are treated in an accessible way, the book is very well suited for an introductory course in number theory.

The following topics are discussed: some elementary number theory (basic facts about primes, unique factorization theorem on \(\mathbb Z\)); Diophantine equations (history on the equations of Fermat and Catalan); quadratic congruences and quadratic equations, unique prime ideal factorization in quadratic number fields; some basics of elliptic curves and applications to Diophantine problems such as the congruent number problem, the taxicab problem and elliptic divisibility sequences; elliptic functions; heights on elliptic curves and a proof of the Mordell Theorem in a special case; proof of the existence of the Néron-Tate height; some basics of the Riemann zeta-function and a deduction of its functional equation; Dirichlet characters, L-functions, and a proof that each arithmetic progression contains infinitely many primes; connections between analytic and algebraic number theory, such as the Dirichlet class number formula, signs for Gauss sums, and the Birch and Swinnerton-Dyer conjecture (a weak form is discussed); some elementary primality tests and factorization methods and a brief discussion on the RSA-public key cryptosystem.

In my opinion, because so many topics are treated in an accessible way, the book is very well suited for an introductory course in number theory.

The following topics are discussed: some elementary number theory (basic facts about primes, unique factorization theorem on \(\mathbb Z\)); Diophantine equations (history on the equations of Fermat and Catalan); quadratic congruences and quadratic equations, unique prime ideal factorization in quadratic number fields; some basics of elliptic curves and applications to Diophantine problems such as the congruent number problem, the taxicab problem and elliptic divisibility sequences; elliptic functions; heights on elliptic curves and a proof of the Mordell Theorem in a special case; proof of the existence of the Néron-Tate height; some basics of the Riemann zeta-function and a deduction of its functional equation; Dirichlet characters, L-functions, and a proof that each arithmetic progression contains infinitely many primes; connections between analytic and algebraic number theory, such as the Dirichlet class number formula, signs for Gauss sums, and the Birch and Swinnerton-Dyer conjecture (a weak form is discussed); some elementary primality tests and factorization methods and a brief discussion on the RSA-public key cryptosystem.

Reviewer: Jan-Hendrik Evertse (Leiden)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11A41 | Primes |

11Dxx | Diophantine equations |

11G05 | Elliptic curves over global fields |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11R11 | Quadratic extensions |

11Y05 | Factorization |

11Y11 | Primality |

11Y16 | Number-theoretic algorithms; complexity |