On characterization of prime divisors of the index of a quadrinomial

TL;DR

The paper addresses the problem of characterizing primes $p$ that divide the discriminant $D$ of a quadrinomial minimal polynomial $f(x)=x^{n}+ax^{n-1}+bx+c$ but do not divide the index $[\mathcal{O}_{K}:\mathbb{Z}[\theta]]$, for $n>4$, $abc\neq 0$, and $n^{2}=ak$. Using Dedekind's criterion and the $M(x)$ construction, it derives a comprehensive set of explicit congruence conditions (across eight cases) that determine when $p\nmid [\mathcal{O}_{K}:\mathbb{Z}[\theta]]$, thereby yielding necessary and sufficient criteria for $\mathbb{Z}[\theta]$ to be integrally closed and enabling monogenity tests for the field $K=\mathbb{Q}(\theta)$. The findings also relate the index and discriminant via $D= [\mathcal{O}_{K}:\mathbb{Z}[\theta]]^{2}D_{K}$ and provide corollaries and propositions describing when primes divide or do not divide $D_{K}$. The paper includes concrete examples that illustrate both monogenic and non-monogenic cases, and a large-degree example that demonstrates the practical application of the criterion to bound primes dividing the discriminant. Collectively, the work advances explicit monogenity criteria for number fields generated by quadrinomials and clarifies how discriminant data interacts with the ring of integers.

Abstract

Let $θ$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $θ$ over the rationals. 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 $f(x).$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(θ),$ where $θ$ is associated with certain quadrinomials.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 1.4
• Proposition 1.5
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4