Isomorphisms of Galois groups of number fields with restricted ramification

Ryoji Shimizu

Mathematische Nachrichten, 2023, vol. 296, issue 7, 3026-3033

Abstract: Let K be a number field and S a set of primes of K. We write KS/K$K_S/K$ for the maximal extension of K unramified outside S and GK,S$G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For i=1,2$i=1,2$, let Ki$K_i$ be a number field, Si$S_i$ a (sufficiently large) set of primes of Ki$K_i$ and σ:GK1,S1→∼GK2,S2$\sigma :G_{K_1,S_1} {\overset{\sim }{\rightarrow }} G_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between K1,S1/K1$K_{1,S_1}/K_1$ and K2,S2/K2$K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of Si$S_i$.

Date: 2023
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