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

Abstract

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

23 pages