Finite Fields

In this report, we revised some important definitions with examples and results of ring theory such as ring homomorphism, Euclidean domain, principal ideal domain, unique factorization domain, polynomial rings, irreducibility criteria etc. Then, we discuss field theory. In field theory, we study the details of extension of fields, splitting fields, algebraic extensions etc. The most important field of abstract algebra is Galois theory. Here, we prove the fundamental theorem of Galois theory and the application of this result. Lastly, we discuss the structure and applications of finite fields