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October 16, 2025Open Access

Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

Key Points

Monogenic quartic polynomials exhibit unique Galois group characteristics based on parameter choices.
For $A e 0$ and $B=0$, specific Galois groups lead to particular monogenic polynomial forms.
Research identifies irreducibility conditions of polynomials using algebraic structures related to Galois theory.
This analysis contributes to our understanding of the relationships between polynomial forms and their Galois groups.

Abstract

Suppose that f (x) =x⁴+Ax³+Bx²+Ax+1 Zx. We say that f (x) is monogenic if f (x) is irreducible over Q and \1, , ², ³\ is a basis for the ring of integers of Q (), where f () =0. For each possible Galois group G that can occur in the two cases of A 0 with B=0, and AB 0, we determine all monogenic polynomials f (x) with Galois group G.

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Lenny Jones (Mon,) studied this question.

synapsesocial.com/papers/68f0d5eb105731330a2b1f9b https://doi.org/https://doi.org/10.48550/arxiv.2502.17691

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