Monogenic Even Cyclic Sextic Polynomials
Lenny Jones
TL;DR
This work classifies monogenicity among even sextic polynomials with particular Galois structures. Using discriminant criteria, Dedekind index tests, and AJ-type conditions for Galois groups, it rules out monogenicity for all monic even cyclic sextic binomials and trinomials, and provides a precise monogenic characterization for three infinite quadrinomial families, yielding exactly four monogenic examples with distinct splitting fields. It further shows that noncyclic cases can exhibit different behavior, constructing an infinite family of monogenic $A_4$-type sextics and giving a clear criterion (squarefree discriminant component $\mathcal{D}$) for monogenicity. Overall, the paper clarifies how monogenicity interacts with the cyclic versus noncyclic (e.g., $A_4$) Galois structures in sextic fields and supplies explicit infinite families for further study.
Abstract
Suppose that $f(x)\in {\mathbb Z}[x]$ is monic and irreducible over ${\mathbb Q}$ of degree $N$. We say that $f(x)$ is monogenic if $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$, and we say $f(x)$ is cyclic if the Galois group of $f(x)$ over ${\mathbb Q}$ is isomorphic to the cyclic group of order $N$. In this note, we prove that there do not exist any monogenic even cyclic sextic binomials or trinomials. Although the complete story on monogenic even cyclic sextic quadrinomials remains somewhat of a mystery, we nevertheless determine that the union of three particular infinite sets of cyclic sextic quadrinomials contains exactly four quadrinomials that are monogenic with distinct splitting fields. We also show that the situation can be quite different for quadrinomials whose Galois group is not cyclic.