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

TL;DR

The paper studies how primes split in nonabelian function-field extensions of Heisenberg type, building on Hirano–Morishita's mod-\ell Heisenberg extensions. It develops an explicit, Euler-criterion–like criterion for complete splitting of the principal ideal $(t-a)$ in \mathcal{R}^{(\ell)}/\mathbb{F}_p(t)$ using a polynomial $A_\ell(a)$, under the assumption that $a$ and $1-a$ are $\ell$-th power residues in $\mathbb{F}_p$ (with a separate treatment for $\ell=2$). The authors construct and analyze an explicit integral basis for \mathcal{R}^{(\ell)}$, compute discriminants, and show that splitting is governed by the value of $A_\ell(a)$, thereby extending a classical Euler criterion to a nonabelian, function-field setting. Collectively, the results link the group-theoretic structure of the mod-\ell Heisenberg extension to concrete arithmetic criteria for prime decomposition, with potential connections to Massey products and Milnor invariants in Galois cohomology.

Abstract

For primes $p$ and $\ell$ such that $\ell$ divides $p-1$, Hirano and Morishita constructed a nonabelian Galois extension of the function field $\mathbb{F}_p(t)$ whose degree is $\ell^3$ 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 $a \in \mathbb{F}_p-\{0,1\}$ 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-$\ell$ Heisenberg group and the arithmetic of field extensions.

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof