A note on monogenic even polynomials

TL;DR

The paper addresses the monogenicity of even sextic polynomials with prescribed Galois groups, focusing on the cyclic group $C_6$. It develops a framework combining resolvent analysis, ramification via $p$-Eisenstein, discriminant considerations, and group-theoretic constraints to classify monogenic instances, yielding a precise finite list of monogenic polynomials in the shape $f(X)=X^6+(n^2/c-2m)X^4+(m^2-2n)X^2+c$ that realize $\mathrm{Gal}(f/\mathbb{Q})\cong C_6$, and a higher-degree analogue for $f(X)=g(X^2)$ with primes $q\ge 5$. The work extends Jones's conjecture to the full family of even cyclic sextics and maps out monogenicity patterns across other Galois groups by employing discriminant analysis, ramification constraints, and Hilbert irreducibility, identifying $C_6$ and $S_3$ as the only groups without infinite monogenic families. It also provides explicit infinite families for other groups (e.g., $6T8$ and certain $S_4$-types) and highlights nonexistence results for dihedral cases $D_m$ with odd $m$, thereby offering a comprehensive landscape of monogenicity among even sextics and related $f(X^\ell)$ constructions.

Abstract

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials with cyclic Galois group. This result is key to completing previous partial results on existence or non-existence of infinite families of even sextic monogenic polynomials with a prescribed Galois group. Some of the underlying ideas are relevant for investigation of more general families of even polynomials $f(X^2)$, or power-compositional polynomials $f(X^\ell)$.

Theorems & Definitions (21)

• Theorem 1.1
• Lemma 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 3.1