On characterization of prime divisors of the index of a quadrinomial

Let $\theta$ be an algebraic integer and $f(x)=x^n+a{x}^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta )$ be a number field and ${\mathcal{O}}_K$ be the ring of integers of $K\mathrm{.}$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x)\mathrm{.}$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(\theta ),$ where $\theta$ is associated with certain quadrinomials.