Monogenic trinomials and class numbers of related quadratic fields
Lenny Jones
TL;DR
The paper investigates monogenic trinomials $f(x)=x^{N}+Ax+B$ whose discriminants $\Delta(f)$ possess divisors matching discriminants of quadratic fields ${\mathbb Q}(\sqrt{\delta})$ (with $\delta\neq \pm 1$ squarefree). By combining Swan’s discriminant formula, the Jakhar–Khanduja–Sangwan monogenicity criteria, and known class-number divisibility results (Murty, Kishi–Miyake, Louboutin), it constructs infinite families of monogenic trinomials whose associated quadratic fields have prescribed divisibility properties of their class numbers, and proves that no two polynomials in each family generate the same extension. The paper provides explicit characterizations of when monogenicity holds in several parametric families (notably $f_{u,w}$, $f_{a,b}$, and $f_{N,b}$ variants), proves irreducibility criteria, and establishes a link between monogenicity and class-number divisibility through concrete theorems. Computational examples, generated with Sage, illustrate the theoretical results and demonstrate the richness of the constructed families, including tables of discriminants, Galois groups, and class numbers. Overall, the work sheds light on the relationship between monogenicity of trinomials and arithmetic properties of associated quadratic fields, and confirms the existence of infinitely many such polynomials with prescribed class-number divisibility behavior.
Abstract
We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we investigate the divisibility of the class numbers of quadratic fields ${\mathbb Q}(\sqrtδ)$ for certain families of monogenic trinomials $f(x)=x^N+Ax+B$, where $δ\ne \pm 1$ is a squarefree divisor of the discriminant of $f(x)$.