The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion

2025/11/21 by Dohyeong Kim, Kim, Dohyeong, Yang, Ingyu
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2511.17095

Abstract

For primes p and ℓ such that ℓ divides p-1, Hirano and Morishita constructed a nonabelian Galois extension of the function field \mathbbF_p(t) whose degree is ℓ³ and Galois group is of Heisenberg type. Here we analyze how primes of degree one decompose in such extensions. It amounts to investigating the decomposition of the principal ideal (t-a) for a ∈ \mathbbF_p-\0,1\ and our main result determines when it decomposes completely in terms of an explicit polynomial in a. It is reminiscent of Euler's criterion. The proof relies on both the group structure of the mod-ℓ Heisenberg group and the arithmetic of field extensions.

Citations

• On Triple Quadratic Residue Symbols in Real Quadratic Fields
• Arithmetic topology in Ihara theory
• On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field
• On a certain nilpotent extension over Q of degree 64 and the 4-th multiple residue symbol

Related

• Hasse norm principle for Heisenberg extensions of degree p³
• Arithmetic equivalence for non-geometric extensions of global function fields
• Skew Braces and Hopf-Galois Structures of Heisenberg Type
• On the local behaviour of specializations of function field extensions
• Idéaux premiers totalement décomposés et sommes de Newton