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

Let ZK denote the ring of algebraic integers of an algebraic number field K = Q(θ ) , where θ 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, θ , … , θ ⁿ⁻¹\ is a basis for ZK . 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[θ ] in ZK . 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.