The Irreducibility and Monogenicity of Power-Compositional Trinomials

2022/04/16 by Joshua Harrington, Lenny Jones, Harrington, Joshua +1
Mathematics · #12F05 #Advanced Differential Equations and Dynamical Systems #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary 11R04 #Secondary 11R09

paper · pdf · doi:10.48550/arxiv.2204.07784

openalex publication_date 2022/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A polynomial f(x)∈ \mathbb Z[x] of degree N is called monogenic if f(x) is irreducible over \mathbb Q and \1,θ,θ²,… ,θ^N-1\ is a basis for the ring of integers of \mathbb Q(θ), where f(θ)=0. Define \mathcal F(x):=x^m+Ax^m-1+B. In this article, we determine sets of conditions on m, A, and B, such that the power-compositional trinomial \mathcal F(x^pn) is monogenic for all integers n≥ 0 and a given prime p. Furthermore, we prove the actual existence of infinite families of such trinomials \mathcal F(x).

Related

• Monogenic cyclic trinomials of the form x⁴+cx+d
• Monogenic trinomials and class numbers of related quadratic fields
• The Monogenicity of Power-Compositional Characteristic Polynomials
• A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and k-Wall-Sun-Sun Primes
• Monogenic Cyclic Quartic Trinomials