2024/04/27 by Lenny Jones, Jones, Lenny · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2404.17921
openalex publication_date 2024/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A monic polynomial f(x)∈ \mathbb Z[x] of degree N is called monogenic if f(x) is irreducible over \mathbb Q and \1,θ,θ2,… ,θN-1\ is a basis for the ring of integers of \mathbb Q(θ), where f(θ)=0. In a series of recent articles, complete classifications of the Galois groups were given for irreducible polynomials \mathcal F(x):=x8+ax4+b∈ \mathbb Z[x] and \mathcal G(x):=x8+ax6+bx4+ax2+1∈ \mathbb Z[x], a≠ 0. In this article, for each Galois group G arising in these classifications, we either construct an infinite family of monogenic octic polynomials \mathcal F(x) or \mathcal G(x) having Galois group G, or we prove that at most a finite such family exists. In the finite family situations, we determine all such polynomials. Here, a ``family" means that no two polynomials in the family generate isomorphic octic fields.