We say that a monic irreducible polynomial \(f(x)\in {\mathbb {Z}[x]}\) is monogenic if for any root \(\theta \) of f(x), \(\mathbb {Z}[\theta ]\) is equal to the ring of integers of the number field \(\mathbb {Q}(\theta )\) . Let \(f(x)=x^n+ax+b\in {\mathbb {Z}[x]}\) be an irreducible trinomial and let \(g(x)\in {\mathbb {Z}[x]}\) be such that \(f\circ g(x)\) is irreducible over \(\mathbb {Q}.\) In this article, we establish a criterion which characterizes the primes dividing the index of \(\mathbb {Z}[\alpha ]\) in the ring of integers of \(K=\mathbb {Q}(\alpha )\) , where \(\alpha \) is a root of \(f\circ g(x)\) . In fact, for certain choices of g(x), we prove that no prime dividing disc(f(x)) divides \([\mathcal {O}_K:\mathbb {Z}[\alpha ]]\) if and only if f(x) is monogenic. We apply this result to find a couple of infinite families of monogenic polynomials over \(\mathbb {Z}\) .
