We associate a positive integer n and a subgroup H of the group (Z/nZ)^x with a polynomial Jn, H(x), which is called the Galois polynomial. It turns out that Jn, H(x) is a polynomial with integer coefficients for any n and H. In this paper, we provide an equivalent condition for a subgroup H to provide the Galois polynomial which is irreducible over Q in the case of n = p1^e1 ···p_r ^er (prime decomposition) with all ei □ 2. For a positive integer n, we denote the n^th primitive root e of unity by □n, and the (multiplicative) group consisting of all invertible elements in the ring Z/nZ by (Z/nZ)^x throughout this paper. Also, □(n) denotes the Euler's phi function, i.e., □(n) =□(Z/nZ)^x□.