On the Discriminant and Index of a Certain Class of Polynomials

Let f(x) = (x²+1)ⁿ - axⁿ ∈ Z[x] and assume $f(x)$ is irreducible. Let θ be a root of $f(x)$ , set K= Q(θ ) and denote by ZK the ring of integers of K . The index of f , denoted ind(f) , is the index of Z[θ ] in ZK . A polynomial $f(x)$ is said to be monogenic if ind(f) = 1 . We compute the discriminant of the polynomial $f(x)$ , and then derive necessary and sufficient conditions on the parameters a and n for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide ind(f) .