Number Theory

Résumé du livre

Number Theory: An Introduction to Mathematics by W.A. CoppelUndergraduate courses in mathematics are commonly of two types. On the one handthere are courses in subjects, such as linear algebra or real analysis, with which it isconsidered that every student of mathematics should be acquainted. On the other handthere are courses given by lecturers in their own areas of specialization, which areintended to serve as a preparation for research. There are, I believe, several reasonswhy students need more than this.First, although the vast extent of mathematics today makes it impossible for anyindividual to have a deep knowledge of more than a small part, it is important to havesome understanding and appreciation of the work of others. Indeed the sometimessurprising interrelationships and analogies between different branches of mathematicsare both the basis for many of its applications and the stimulus for further develop-ment. Secondly, different branches of mathematics appeal in different ways and requiredifferent talents. It is unlikely that all students at one university will have the sameinterests and aptitudes as their lecturers. Rather, they will only discover what theirown interests and aptitudes are by being exposed to a broader range. Thirdly, manystudents of mathematics will become, not professional mathematicians, but scientists,engineers or schoolteachers. It is useful for them to have a clear understanding of thenature and extent of mathematics, and it is in the interests of mathematicians that thereshould be a body of people in the community who have this understanding.The present book attempts to provide such an understanding of the nature andextent of mathematics. The connecting theme is the theory of numbers, at first sightone of the most abstruse and irrelevant branches of mathematics. Yet by exploringits many connections with other branches, we may obtain a broad picture. The topicschosen are not trivial and demand some effort on the part of the reader. As Euclidalready said, there is no royal road. In general I have concentrated attention on thosehard-won results which illuminate a wide area. If I am accused of picking the eyes outof some subjects, I have no defence except to say "But what beautiful eyes!"The book is divided into two parts. Part A, which deals with elementary numbertheory, should be accessible to a first-year undergraduate. To provide a foundation forsubsequent work, Chapter I contains the definitions and basic properties of variousmathematical structures. However, the reader may simply skim through this chapter and refer back to it later as required. Chapter V, on Hadamard's determinant problem,shows that elementary number theory may have unexpected applications.Part B, which is more advanced, is intended to provide an undergraduate with someidea of the scope of mathematics today. The chapters in this part are largely indepen-dent, except that Chapter X depends on Chapter IX and Chapter XIII on Chapter XII.