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arXiv · 2107.04280

Isomorphisms of Galois groups of number fields with restricted ramification

Ryoji Shimizu
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Abstract

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: "For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\to G_{K_2,S_2}$ an isomorphism. Is $\sigma$ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?" Here the main assumption is about the Dirichlet density of $S_i$.

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Ryoji Shimizu. 2021-07-09. Isomorphisms of Galois groups of number fields with restricted ramification. https://arxiv.org/abs/2107.04280

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