This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. It discusses several combinatorial applications of finite fields such as Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.
Primes and factorization Primes in the integers Congruences in rings Primes in polynomial rings: Constructing finite fields The additive and multiplicative structures of finite fields Understanding the structures of Z/nZ Combinatorial applications of finite fields The AKS primality test Synopsis of finite fields