Small Galois groups that encode valuations

TL;DR

This paper demonstrates that a small canonical Galois group over a field containing a p-th root of unity encodes specific valuations on the field, revealing new insights into the relationship between Galois groups and valuation theory.

Contribution

It introduces a minimal Galois group that encodes valuations with certain properties, advancing understanding of Galois groups' role in valuation theory.

Findings

The group $(G_F)_{[3]}$ encodes valuations with non-$p$-divisible value groups.

Valuations satisfying a variant of Hensel's lemma are characterized by this Galois group.

The work links small Galois groups to valuation-theoretic properties in fields containing roots of unity.

Abstract

Let $p$ be a prime number and let $F$ be a field containing a root of unity of order $p$. We prove that a certain very small canonical Galois group $(G_F)_{[3]}$ over $F$ encodes the valuations on $F$ whose value group is not $p$-divisible and which satisfy a variant of Hensel's lemma.