Polynomial functions over finite fields are a major tool in computer science and electrical engineering and have a long history. Some of its aspects, like interpolation and permutation polynomials are described in this thesis. A complete characterization of subfield compatible polynomials (f in E[x] such that f(K) is a subset of L, where K,L are subfields of E) was recently given by J. Hull. In his work, he introduced the Frobenius permutation which played an important role. In this thesis, we fully describe the cycle structure of the Frobenius permutation. We generalize it to a permutation called a monomial permutation and describe its cycle factorization. We also derive some important congruences from number theory as corollaries to our work.