I will introduce the basics of Fourier analysis on finite Abelian groups and present some of its applications to number theory. In particular, I will discuss Dirichlet’s theorem. That is, when a and d are coprime, there are infinitely many primes of the form a+nd, where n is a nonnegative integer. Several examples illustrating this theorem will be presented. The talk should be accessible to upper-level undergraduates and above.