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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2408.12441 |
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| _version_ | 1866917767160004608 |
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| author | Entin, Alexei |
| author_facet | Entin, Alexei |
| contents | We study the following question: given a global field $F$ and finite group $G$, what is the minimal $r$ such that there exists a finite extension $K/F$ with $\mathrm{Aut}(K/F)\cong G$ that is ramified over exactly $r$ places of $F$? We conjecture that the answer is $\le 1$ for any global field $F$ and finite group $G$. In the case when $F$ is a number field we show that the answer is always $\le 4[F:\mathbb Q]$. We show that assuming Schinzel's Hypothesis H the answer is always $\le 1$ if $F$ is a number field. We show unconditionally that the answer is always $\le 1$ if $F$ is a global function field.
We also show that for a broader class of fields $F$ than previously known, every finite group $G$ can be realized as the automorphism group of a finite extension $K/F$ (without restriction on the ramification).
An important new tool used in this work is a recent result of the author and C. Tsang, which says that for any finite group $G$ there exists a natural number $n$ and a subgroup $H\leqslant S_n$ of the symmetric group such that $N_{S_n}(H)/H\cong G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphism Groups of Finite Extensions of Fields and the Minimal Ramification Problem Entin, Alexei Number Theory We study the following question: given a global field $F$ and finite group $G$, what is the minimal $r$ such that there exists a finite extension $K/F$ with $\mathrm{Aut}(K/F)\cong G$ that is ramified over exactly $r$ places of $F$? We conjecture that the answer is $\le 1$ for any global field $F$ and finite group $G$. In the case when $F$ is a number field we show that the answer is always $\le 4[F:\mathbb Q]$. We show that assuming Schinzel's Hypothesis H the answer is always $\le 1$ if $F$ is a number field. We show unconditionally that the answer is always $\le 1$ if $F$ is a global function field. We also show that for a broader class of fields $F$ than previously known, every finite group $G$ can be realized as the automorphism group of a finite extension $K/F$ (without restriction on the ramification). An important new tool used in this work is a recent result of the author and C. Tsang, which says that for any finite group $G$ there exists a natural number $n$ and a subgroup $H\leqslant S_n$ of the symmetric group such that $N_{S_n}(H)/H\cong G$. |
| title | Automorphism Groups of Finite Extensions of Fields and the Minimal Ramification Problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.12441 |