Number Theory and Its Applications

This book emphasizes the role of symmetry and presents as many viewpoints as possible of an important phenomenon — the functional equation of the associated zeta-function. It starts from the basics before warping into the space of new interest; from the ground state to the excited state. For example, the Euler function is treated in several different places, as the number of generators of a finite cyclic group, as one counting the order of the multiplicative group of reduced residue classes modulo q, and as the order and degree of the Galois group of the cyclotomic field, respectively.One of the important principles of learning is to work with the material many times. This book presents many worked-out examples and exercises to enhance the reader's comprehension on the topics covered in an in-depth manner. This is done in a differ-ent setting each time such that the reader will always be challenged. For the keen reader, even browsing the text alone, without solving the exercises, will yield some knowledge and enjoyment.