Construction of Finite Fields

The theory of Finite fields plays a significant role in the theory of Galois extensions. Finite fields have many applications in Coding theory, Computing and Statistics. Therefore, this paper makes an attempt to study some finite fields and their properties. Especially it concentrates on the construction of finite fields. It is known that the characteristic of a finite field is a prime number. Here, first we construct the finite fields containing 4, 8, and 16 elements of characteristic 2, then the fields with 9 and 27 elements of characteristic 3, and finally the fields of 25 elements with characteristic 5. It is well known that the multiplicative group of a finite field is a cyclic group. Therefore, the generators of all the cyclic groups of the finite fields are also found.