Open Homomorphisms between $m$-step Solvable Galois Groups Compatible with the Cyclotomic Characters

TL;DR

This paper extends Hoshi's result by proving that open homomorphisms between certain solvable Galois groups of number fields, compatible with cyclotomic characters, originate from field embeddings, specifically for $m$-step solvable groups.

Contribution

It establishes an $m$-step solvable version of Hoshi's theorem, linking open homomorphisms and field embeddings under cyclotomic character compatibility.

Findings

Open homomorphisms between $m+3$-step solvable Galois groups correspond to field embeddings.

Compatibility with cyclotomic characters is necessary and sufficient for homomorphisms to arise from field embeddings.

The result generalizes Hoshi's theorem to a broader class of solvable Galois groups.

Abstract

In \cite{Ho3}, Hoshi proved that open homomorphisms between solvably closed Galois groups of number fields which are compatible with the cyclotomic characters arise from field embeddings. In this paper, we will prove an $m$-step solvable version of Hoshi's result. More precisely, if $K$ and $L$ are number fields, we will prove that given an open homomorphism between the maximal $m+3$-step solvable Galois groups of $K$ and $L$, where $m \geq 2$, and the induced open homomorphism between the corresponding maximal $m$-step solvable Galois groups, then the latter arises from a field embedding if and only if the open homomorphism between the $m+3$-step solvable (and hence also the $m$-step solvable) Galois groups is compatible with the cyclotomic characters of $K$ and $L$.