Isomorphisms of Galois groups
Kôji Uchida · Journal of the Mathematical Society of Japan · 1976
Let $Q$ be the field of the rational numbers.Let $\Omega$ be a normal algebraic extension of $Q$ such that $\Omega$ has no abelian extension.Let $G$ be the Galois group of $\Omega$ over $Q$ .Neukirch $[4, 5]$ has shown that every open normal sub- group of $G$ is a characteristic subgroup, and has proposed a problem whether every automorphism of $G$ is inner.In this paper this problem is solved affir- matively, $i$ .$e.$ , we prove THEOREM.Let $G_{1}$ and $G_{2}$ be open subgroups of $G$ , and let $\sigma;G_{1}\rightarrow G_{2}$ be a topological isomorphism.Then $\sigma$ can be extended to an inner automorPhism of $G$ .Kanno [2] and Komatsu [3] gave partial results which suggested this problem is affirmative.The author first proved this theorem in the case $\Omega$ is the algebraic closure or the solvable closure of $Q$ , because Neukirch has stated his theorems in these cases.Ikeda [1] solved Neukirch's problem independ- ently almost at the same time.Iwasawa then remarked that his methods are also applicable to the proof of our theorem.Iwasawa also noticed that Neu- kirch's theorems are valid for every $\Omega$ as above.We state our theorem