Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to S_p or A_p.