ON THE DISCRIMINANT AND INDEX OF A CERTAIN CLASS OF POLYNOMIALS

Abstract Let $f(x) = (x^{2}+1)^{n} - ax^{n} \in \mathbb {Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta $ be a root of $f(x)$ , set $K= \mathbb {Q}(\theta )$ and denote by $\mathbb {Z}_{K}$ the ring of integers of K . The index of f , denoted $\operatorname {ind}(f)$ , is the index of $\mathbb {Z}[\theta ]$ in $\mathbb {Z}_{K}$ . A polynomial $f(x)$ is said to be monogenic if $\operatorname {ind}(f) = 1$ . 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 $\operatorname {ind}(f)$ .