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

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.