Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.17095 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914387323781120 |
|---|---|
| author | Kim, Dohyeong Yang, Ingyu |
| author_facet | Kim, Dohyeong Yang, Ingyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion Kim, Dohyeong Yang, Ingyu Number Theory 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. |
| title | The decomposition of primes in nonabelian extensions of Heisenberg type and an analogue of Euler's criterion |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.17095 |