Monogenic Reciprocal Quartic Polynomials And Their Galois Groups
Lenny Jones
TL;DR
This work classifies monogenic, reciprocal quartic polynomials $f(x)=x^4+Ax^3+Bx^2+Ax+1$ over ${\mathbb Z}$ according to their Galois groups in two regimes: $A\ne 0$, $B=0$ and $AB\ne 0$. It uses a three-stage framework: determine irreducibility, apply Dedekind's index criterion to test when $\mathbb{Z}[\theta]$ is the full ring of integers, and then classify Galois groups via discriminant invariants. In the $B=0$ case (with $A\notin\{0,\pm1\}$), monogenicity occurs exactly when $2\mid A$ and $A-1$, $A+1$, and $(A/2)^2+2$ are squarefree, yielding an infinite family with ${\rm Gal}(f)\simeq D_4$, and a complete distinct-subfamily ${\mathcal F}^{+}$ for $A\ge 2$. In the $AB\ne 0$ case, monogenic polynomials partition into five families; four are infinite with ${\rm Gal}(f)\simeq D_4$, while a finite six-polynomial set ${\mathcal F}_5$ has ${\rm Gal}(f)\simeq C_4$. The work provides explicit constructions and discriminant-based criteria linking squarefreeness conditions to both monogenicity and Galois structure, enabling precise identification of monogenic number fields arising from reciprocal quartics.
Abstract
Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,θ^3\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. For each possible Galois group $G$ that can occur in the two cases of $A\ne 0$ with $B=0$, and $AB\ne 0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$.