Monogenic even sextic trinomials and their Galois groups

Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb Z}[x]$, with $A\ne 0$ and $k\in {1,2}$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and ${1,θ,θ^2,θ^3,θ^4,θ^{5}}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. For each value of $k$ and each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also determine when these descriptions provide infinitely many such trinomials, and we investigate when these trinomials generate distinct sextic fields. These results extend recent work on monogenic power-compositional sextic trinomials of the form $g(x^3)$ to the situation $g(x^2)$, and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials.

💡 Research Summary

This paper presents a complete characterization of monogenic even sextic trinomials of the form f(x) = x^6 + Ax^(2k) + B, where A and B are nonzero integers and k ∈ {1, 2}. A polynomial is defined as monogenic if it is irreducible over the rationals Q and the set {1, θ, θ^2, θ^3, θ^4, θ^5} forms an integral basis (a power basis) for the ring of integers of the field Q(θ), where f(θ)=0. The primary goal is to describe explicitly, for each possible Galois group G of f(x) over Q, all monogenic trinomials f(x) having Galois group isomorphic to G. Furthermore, the paper investigates when these descriptions yield infinitely many such trinomials and determines conditions under which these trinomials generate distinct sextic fields.

The work extends recent research on monogenic power-compositional sextic trinomials of the form g(x^3) to those of the form g(x^2), thereby completing the Galois group classification for all monogenic power-compositional sextic trinomials. The key auxiliary polynomials are g(x) = x^3 + Ax^k + B, h(x) = x^6 + (-1)^k AB^(k-1)x^(6-2k) - B^2, and δ = 4A^3 + 27B^(3-k). Their discriminants are derived as Δ(f) = -64B^(2k-1)δ^2 and Δ(g) = -B^(k-1)δ.

The foundational tool for the monogenicity analysis is a theorem of Jakhar, Khanduja, and Sangwan, which provides necessary and sufficient conditions for an arbitrary irreducible trinomial to be monogenic in terms of the prime factors of its discriminant. The author specializes this theorem to the specific form f(x), resulting in Theorem 2.8. This theorem gives precise arithmetic conditions modulo powers of 2, 3, and other primes that the coefficients A and B must satisfy for f(x) to be monogenic, depending on how a prime divisor p of Δ(f) relates to A and B.

The Galois group of f(x) is determined using adapted criteria from prior work, summarized in Theorem 2.5. The group depends on whether -B, Δ(g), and -BΔ(g) are squares in Z, and on the irreducibility of h(x) over Q. For a general even sextic, eight Galois groups are possible (C6, S3, C2×S3, A4, C2×A4, S4+, S4-, C2×S4, using common names and T-nomenclature 6T1-6T11). A crucial lemma (Lemma 3.1) shows that if h(x) is reducible, then Δ(g) < 0, which immediately implies that no monogenic f(x) can have the cyclic Galois group C6 (Corollary 3.2).

The main result, Theorem 1.1, provides necessary and sufficient conditions on A, B, and k for f(x) to be monogenic with each specific Galois group G. The findings are striking in their specificity and asymmetry:

• C6 and S3: No monogenic trinomials exist with these Galois groups.
• C2×S3: Occurs only for k=2 with (A,B) ∈ {(-2,2), (2,-2)}.
• A4: Occurs only for (k,A,B) ∈ {(1,-3,-1), (2,-3,-1)}.
• C2×A4: For k=1, there is an infinite family F1 defined by conditions including B being squarefree, B≠-1, rad(|δ|) dividing A, and specific congruences modulo 9 if 3 divides A. For k=2, only three isolated pairs (A,B) work.
• S4+: Characterized by an infinite family F2, with core conditions B=-1, A not equal to (-1)^k*3, A not divisible by 4 or 9, and δ/3^(ν₃(δ)) being squarefree.
• S4-: For k=1, only (A,B)=(-9,-6). For k=2, an infinite family F3 exists, requiring 3|A, 4∤A, B≠-1, and B = 3 - 4(A/3)^3 being squarefree.
• C2×S4: Defined by the most complex set of conditions, involving squarefreeness of B and a modified δ, non-square status of -B^(k-1)δ, specific mod 4 conditions, and mod 9 conditions (sets R1 or R2) when 3|A and 3∤B.

Corollary 1.2 demonstrates that this classification is not merely a finite list. The families F1, F2, and F3 for the groups C2×A4, S4+, and S4- are all infinite. Moreover, each contains at least one infinite one-parameter subfamily whose members generate pairwise distinct sextic fields (a fact often proven by showing their discriminants are distinct). Perhaps more significantly, the corollary also constructs, for each k ∈ {1,2}, an infinite one-parameter family of monogenic trinomials with Galois group C2×S4, again with all members generating distinct fields.

The paper’s methodology exemplifies a powerful synergy of algebraic number theory and Galois theory. By first constraining the algebraic structure (the Galois group) and then applying precise arithmetic criteria for monogenicity, the author achieves a exhaustive classification for a non-trivial class of polynomials. The construction of infinite parametric families shows that monogenicity, while a restrictive property, occurs in rich and systematic patterns within this class, moving beyond sporadic examples.