Characterisation of Primes Dividing the Index of a Class of Polynomials and Its Applications

Abstract Let {Z}₊ denote the ring of algebraic integers of an algebraic number field K = Q (), where is a root of a monic irreducible polynomial f (x) = xⁿ + a (bx+c) ᵐ {Z}x, 1 m<n. We say f (x) is monogenic if \1, , , ^{n-1\} is a basis for {Z}K. We give necessary and sufficient conditions involving only a, b, c, m, n for f (x) to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup {Z} in {Z}K. As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group Sₙ, the symmetric group on n letters.