2024/08/22 by Alexei Entin, Entin, Alexei
Mathematics · Medicine · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2408.12441
openalex publication_date 2024/08/22 · openalex created_date 2024/09/29 · openalex updated_date 2026/07/28
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 Aut(K/F)≅ G that is ramified over exactly r places of F? We conjecture that the answer is ≤ 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 ≤ 4[F:\mathbb Q]. We show that assuming Schinzel's Hypothesis H the answer is always ≤ 1 if F is a number field. We show unconditionally that the answer is always ≤ 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 Sn of the symmetric group such that NSn(H)/H≅ G.