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

TL;DR

The paper addresses identifying when the number field $K=\mathbb{Q}(\theta)$, with $\theta$ a root of $f(x)=x^{n}+a x^{3}+b x+c$, is monogenic. Under the constraints $n=3k>4$, $\frac{a}{a-c}=k$ and $2ab=3ac-bc$, the authors apply Dedekind's criterion to classify primes $p$ dividing the discriminant $D_{f}$ that do not divide the index $[\mathcal{O}_{K}:\mathbb{Z}[\theta]]$, yielding explicit necessary and sufficient conditions for monogenicity that depend only on $a,b,c,n$. A central contribution is the formulation of conditions (1)–(6) that characterize when $p \nmid [\mathcal{O}_{K}:\mathbb{Z}[\theta]]$, enabling a complete, prime-wise monogenicity test and leading to a corollary that $\mathcal{O}_{K}=\mathbb{Z}[\theta]$ under these conditions. The work also reveals a connection to differential equations through an auxiliary polynomial framework and demonstrates practical validity via illustrative examples. Overall, the results provide concrete, implementable criteria to identify monogenic quadrinomial-associated number fields and illuminate the interplay between discriminants, indices, and the ring of integers.

Abstract

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

Theorems & Definitions (17)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4