Theoretical study reveals necessary and sufficient conditions for monogenicity in certain quadrinomial number fields, highlighting exact criteria for associated differential equations.
Let f(x)=xⁿ+ax³+bx+c be the minimal polynomial of an algebraic integer θ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let K=Q(θ) be a number field and OK 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 θ. As an interesting result, we establish necessary and sufficient conditions for the field K=Q(θ) to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$
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Chatterjee et al. (2024) studied this question.
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