A Class of Trinomials with Galois Group Sn

A well known result of Schur states that if n is a positive integer and a 0 , a 1 ,…,a n are arbitrary integers with a 0 a n coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each a i = 1, it is known that the Galois group of f n (x) over ℚ contains A n , the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials f n (x) which leads to the construction of trinomials of degree n for each n with Galois group S n , the symmetric group on n letters.