Tame Galois Groups, Linking Numbers and Mildness
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| author | Feuerpfeil, Julian Hamza, Oussama Lim, Donghyeok |
| author_facet | Feuerpfeil, Julian Hamza, Oussama Lim, Donghyeok |
| contents | Let $p$ be an odd prime and let $S$ be a set of tame primes. We denote by $G_S$ the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside $S$.
We prove that for every finite set of tame primes $S_0$ with $|S_0|\geq 2$, there exists a set $S_1$ consisting of two tame primes such that $G_{S_0\cup S_1}$ has cohomological dimension $2$. This refines a result of Labute. More generally, we establish an analogous result for number fields not containing a primitive $p$-th root of unity, under a suitable splitting condition.
Our approach answers a question of Labute, from his seminal paper on mild groups, and combines weighted Zassenhaus filtrations, graph-theoretic methods, and Koch-type presentations. As an application, we solve several cohomological Galois inverse problems with prescribed ramification and splitting. We also provide numerical examples and statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01083 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tame Galois Groups, Linking Numbers and Mildness Feuerpfeil, Julian Hamza, Oussama Lim, Donghyeok Number Theory Group Theory 11R32, 20E18, 20F14, 20F05, 20F40, 05C25 Let $p$ be an odd prime and let $S$ be a set of tame primes. We denote by $G_S$ the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside $S$. We prove that for every finite set of tame primes $S_0$ with $|S_0|\geq 2$, there exists a set $S_1$ consisting of two tame primes such that $G_{S_0\cup S_1}$ has cohomological dimension $2$. This refines a result of Labute. More generally, we establish an analogous result for number fields not containing a primitive $p$-th root of unity, under a suitable splitting condition. Our approach answers a question of Labute, from his seminal paper on mild groups, and combines weighted Zassenhaus filtrations, graph-theoretic methods, and Koch-type presentations. As an application, we solve several cohomological Galois inverse problems with prescribed ramification and splitting. We also provide numerical examples and statistics. |
| title | Tame Galois Groups, Linking Numbers and Mildness |
| topic | Number Theory Group Theory 11R32, 20E18, 20F14, 20F05, 20F40, 05C25 |
| url | https://arxiv.org/abs/2606.01083 |