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Let f (x) =x^n+ax^3+bx+c be the minimal polynomial of an algebraic integer over the rationals with certain conditions on a, ~b, ~c, and n. Let K=Q () be a number field and O₊ 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. As an interesting result, we establish necessary and sufficient conditions for the field K=Q () to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial f (x).
Chatterjee et al. (Mon,) studied this question.