On characterization of Monogenic number fields associated with certain quadrinomials and its applications

Let $f(x)=x^n+a{x}^3+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,b,c,$ and $n\mathrm{.}$ 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 $\theta \mathrm{.}$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(\theta )$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x)\mathrm{.}$