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

TL;DR

This paper studies how primes decompose in specific nonabelian Galois extensions of function fields of Heisenberg type, providing explicit criteria for complete decomposition related to an analogue of Euler's criterion.

Contribution

It introduces explicit conditions for prime decomposition in nonabelian Heisenberg-type extensions, linking group structure and arithmetic properties.

Findings

Determines when primes decompose completely in these extensions.

Provides an explicit polynomial criterion for prime decomposition.

Establishes an analogue of Euler's criterion for nonabelian extensions.

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.