Some subgroups of a finite field and their applications for obtaining explicit factors

Abstract

Let S q denote the group of all square elements in the multiplicative group F * q of a finite field F q of odd characteristic containing q elements. Let O q be the set of all odd order elements of F * q . Then O q turns up as a subgroup of S q . In this paper, we show that O q = 4 if q = 2t + 1 and, O q = t if q = 4t + 1 , where q and t are odd primes. This paper also gives a direct method for obtaining the coefficients of irreducible factors of x 2 n t − 1 in F q [x] using the information of generator elements of S q and O q , when q and t are odd primes such that q = 2t + 1 or q = 4t + 1 .