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

Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $\theta.$ 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 $f(x).$