Number theory is a branch of mathematics which draws its vitality from a
rich historical background. It is also traditionally nourished through
interactions with other areas of research, such as algebra, algebraic
geometry, topology, complex analysis and harmonic analysis. More
recently, it has made a spectacular appearance in the field of
theoretical computer science and in questions of communication,
cryptography and error-correcting codes. Providing an elementary
introduction to the central topics in number theory, this book spans
multiple areas of research. The first part corresponds to an advanced
undergraduate course. All of the statements given in this part are of
course accompanied by their proofs, with perhaps the exception of some
results appearing at the end of the chapters. A copious list of
exercises, of varying difficulty, are also included here. The second
part is of a higher level and is relevant for the first year of graduate
school. It contains an introduction to elliptic curves and a chapter
entitled "Developments and Open Problems", which introduces and brings
together various themes oriented toward ongoing mathematical research.
Given the multifaceted nature of number theory, the primary aims of this
book are to: - provide an overview of the various forms of mathematics
useful for studying numbers - demonstrate the necessity of deep and
classical themes such as Gauss sums - highlight the role that arithmetic
plays in modern applied mathematics - include recent proofs such as the
polynomial primality algorithm - approach subjects of contemporary
research such as elliptic curves - illustrate the beauty of arithmetic
The prerequisites for this text are undergraduate level algebra and a
little topology of Rn. It will be of use to undergraduates, graduates
and phd students, and may also appeal to professional mathematicians as
a reference text.
