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arXiv · 2511.17095

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

Dohyeong Kim·Ingyu Yang
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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.

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Dohyeong Kim, Ingyu Yang. 2025-11-21. The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion. https://arxiv.org/abs/2511.17095

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