In this thesis we investigate some number theory problems, with special emphasis on Dirichlet’s theorem on arithmetic progressions. We first use algebraic, and later analytic methods to solve our problems. In the first chapter we describe finite fields, define Legendre symbol and prove quadratic reciprocity law. In the second chapter we deal with \(p\)-adic fields. We study the ring of \(p\)-adic integers, \(\mathbb{Z}_{p}\), and field of its fractions, \(\mathbb{Q}_{p}\), and obtain some important results about the solutions of \(p\)-adic equations, whose coefficients are \(p\)-adic integers. In the third chapter, the objects of our observation are characters of finite abelian groups. We first consider characters in general, and then we concentrate on the characters of \(G(m)\), proving some of their properties. In the fourth, final, chapter, we prove the main goal of this thesis, Dirichlet’s theorem on arithmetic progressions. More specifically, we give a proof of its stronger version, using methods of analytic number theory. We study the zeta function and the \(L\)-functions, and define the density of some subset of the set of prime numbers, and then, using obtained results, we prove the theorem mentioned above.