Monogenic sextic trinomials x⁶+Ax³+B and their Galois groups

2025/07/22 by Joshua Harrington, Lenny Jones, Harrington, Joshua +1 · 1 citation
Mathematics · #Mathematics and Applications #Algebraic and Geometric Analysis #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2507.17021

Abstract

Let f(x)=x⁶+Ax³+B∈ \mathbb Z[x], with A≠ 0, and suppose that f(x) is irreducible over \mathbb Q. We define f(x) to be \em monogenic if \1,θ,θ²,θ³,θ⁴,θ⁵\ is a basis for the ring of integers of \mathbb Q(θ), where f(θ)=0. For 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 investigate when these trinomials generate distinct sextic fields.

Citations

• Monogenic Reciprocal Quartic Polynomials And Their Galois Groups
• Monogenic Even Cyclic Sextic Polynomials
• Monogenic cyclic trinomials of the form x⁴+cx+d
• Monogenic trinomials of the form x⁴+ax³+d and their Galois groups
• An elementary computation of the Galois groups of symmetric sextic trinomials
• Square-free values of reducible polynomials

Cited by

• Calculating generators of power integral bases in sextic fields with a quadratic subfield: the general case

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• On index divisors and monogenity of certain sextic number fields defined by x⁶+ax⁵+b
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• The monogenicity and Galois groups of certain reciprocal quintinomials
• On index divisors and monogenity of certain number fields defined by trinomials