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

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