Hopf-Galois module structure of monogenic orders in cubic number fields

2025/06/14 by Daniel Gil-Muñoz, Gil-Muñoz, Daniel
Mathematics · #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2506.12451

Abstract

For a cubic number field L, we consider the ℤ-order in L of the form ℤ[α], where α is a root of a polynomial of the form x³-ax+b and a,b∈ℤ are integers such that v_p(a)≤ 2 or v_p(b)≤ 3 for all prime numbers p. We characterize the freeness of ℤ[α] as a module over its associated order in the unique Hopf-Galois structure H on L in terms of the solvability of at least one between two generalized Pell equations in terms of a and b. We determine when the equality O_L=ℤ[α] is satisfied in terms of congruence conditions for a and b. For such cases, we specialize our result so as to obtain criteria for the freeness of O_L as a module over its associated order in H.

Related

• Galois module structure of algebraic integers of the simplest cubic field
• Hopf Galois module structure of dihedral degree 2p extensions of ℚ_p
• Galois module structure of algebraic integers of cyclic cubic fields
• Hopf-Galois Structures of Squarefree Degree
• The proportion of monogenic orders of prime power indices of the pure cubic field