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**A course in number theory.**
*(English)*
Zbl 0637.10002

Oxford Science Publications. Oxford (UK): Clarendon Press. xi, 354 p. £17.50 (pbk); £40.00 (hbk) (1988).

This is a nice introductory book in number theory concentrating on the elementary methods. Topics, which use the theory of a complex variable or methods from algebraic number theory, are excluded. The book deals with many of the basic concepts and gives a wide and simple introduction. Four important and difficult problems are discussed in more detail. They are the Gel’fond-Schneider Theorem, the class number formula for quadratic forms, the Prime Number Theorem and the Mordell-Weil Theorem.

The first two chapters are an introduction to the foundations of number theory, to divisibility and multiplicative functions. Chapters 3 and 4 are concerned with congruence theory including the quadratic residues. The quadratic reciprocity law is proved by means of Gauss’ Lemma. Besides of the Legendre symbol also the Jacobi and Kronecker symbols are considered. Chapter 5 deals with algebraic numbers, primitive roots and characters.

In Chapter 6 the theory of Gauss sums is developed and a new proof of the quadratic reciprocity law by means of these sums is given. The sign of the quadratic Gauss sum is computed by a method due to Schur. Chapter 7 gives a very short introduction to continued fractions. Chapter 8 deals with transcendental numbers. Theorems of Liouville, of Hermite (transcendence of e) and of Lindemann (transcendence of \(\pi\) and others) are proved. Finally, the famous theorem of Gel’fond-Schneider is proved.

In Chapter 9 quadratic forms with integers as coefficients are considered. The finiteness theorem for the number of equivalence classes is proved, and it is used to solve the three-squares problem. Finally, an algorithm for finding a reduced form for each equivalence class of binary forms is given. Chapter 10 deals with the class group, that is the set of equivalence classes of binary forms. The formula for the number of classes h(d) for the discriminant d is derived.

The very short Chapter 11 considers only some simple properties of partitions.

Chapters 12 and 13 are concerned with the distribution of prime numbers. It is given an elementary proof of the Prime Number Theorem based on Selberg’s formula and a simplification of Shapiro. Also an elementary proof of Dirichlet’s theorem is presented.

Chapter 14 is on Diophantine equations. It includes Legendre’s theorem, a proof of Fermat’s conjecture for the exponents 3 and 4, a short sketch of Skolem’s method and a few special cases of Mordell’s equation. The last Chapter 15 gives an introduction to the theory of elliptic curves. It finishes with a sketch of the theorem of Mordell-Weil.

There are many interesting exercises and problems at the end of each chapter and hints for their solution in the Appendix. Moreover, each chapter contains many informative notes. The author has produced an excellent textbook for an introductory course in number theory.

The first two chapters are an introduction to the foundations of number theory, to divisibility and multiplicative functions. Chapters 3 and 4 are concerned with congruence theory including the quadratic residues. The quadratic reciprocity law is proved by means of Gauss’ Lemma. Besides of the Legendre symbol also the Jacobi and Kronecker symbols are considered. Chapter 5 deals with algebraic numbers, primitive roots and characters.

In Chapter 6 the theory of Gauss sums is developed and a new proof of the quadratic reciprocity law by means of these sums is given. The sign of the quadratic Gauss sum is computed by a method due to Schur. Chapter 7 gives a very short introduction to continued fractions. Chapter 8 deals with transcendental numbers. Theorems of Liouville, of Hermite (transcendence of e) and of Lindemann (transcendence of \(\pi\) and others) are proved. Finally, the famous theorem of Gel’fond-Schneider is proved.

In Chapter 9 quadratic forms with integers as coefficients are considered. The finiteness theorem for the number of equivalence classes is proved, and it is used to solve the three-squares problem. Finally, an algorithm for finding a reduced form for each equivalence class of binary forms is given. Chapter 10 deals with the class group, that is the set of equivalence classes of binary forms. The formula for the number of classes h(d) for the discriminant d is derived.

The very short Chapter 11 considers only some simple properties of partitions.

Chapters 12 and 13 are concerned with the distribution of prime numbers. It is given an elementary proof of the Prime Number Theorem based on Selberg’s formula and a simplification of Shapiro. Also an elementary proof of Dirichlet’s theorem is presented.

Chapter 14 is on Diophantine equations. It includes Legendre’s theorem, a proof of Fermat’s conjecture for the exponents 3 and 4, a short sketch of Skolem’s method and a few special cases of Mordell’s equation. The last Chapter 15 gives an introduction to the theory of elliptic curves. It finishes with a sketch of the theorem of Mordell-Weil.

There are many interesting exercises and problems at the end of each chapter and hints for their solution in the Appendix. Moreover, each chapter contains many informative notes. The author has produced an excellent textbook for an introductory course in number theory.

Reviewer: E.KrĂ¤tzel

### MSC:

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

11Axx | Elementary number theory |

11Dxx | Diophantine equations |

11Exx | Forms and linear algebraic groups |

11Jxx | Diophantine approximation, transcendental number theory |

11Lxx | Exponential sums and character sums |

11N05 | Distribution of primes |