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October 8, 2025Open Access

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

Key Points

The study identifies specific monogenic trinomials with defined Galois groups, which enhances understanding of their algebraic properties.
By applying the theorem of Jakhar, Khanduja, and Sangwan, the research provides explicit descriptions of monogenic sextic trinomials with various Galois groups.
The distinctness of sextic fields generated by these trinomials is investigated, emphasizing the significance of Galois group classifications.
Understanding these relationships may lead to new insights in the fields of algebra and number theory, particularly in the context of irreducible polynomials.

Abstract

Let f (x) =x⁶+Ax³+B Zx, with A 0, and suppose that f (x) is irreducible over Q. We define f (x) to be monogenic if \1, θ, θ², θ³, θ⁴, θ^{5\} is a basis for the ring of integers of Q (θ), where f (θ) =0. For each possible Galois group G of f (x) over 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.

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Cite this study

Harrington et al. (Tue,) studied this question.

www.synapsesocial.com/papers/68e6494525bc5bdb98713b35 — DOI: https://doi.org/10.48550/arxiv.2507.17021

Authors

Joshua Harrington

Lenny Jones

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

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Abstract

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Discussion

Cite this study

Authors

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References and Citations

Citation Network

Connected Papers

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