The tame-wild principle for discriminant relations for number fields

TL;DR

This paper investigates how discriminant divisibility relations, established under tame ramification conditions, often extend to wild ramification cases in number field resolvent constructions, using group-theoretic methods.

Contribution

It demonstrates that many discriminant divisibility relations remain valid in wild ramification scenarios, extending the tame-wild principle in number field theory.

Findings

Discriminant divisibility relations hold under tame ramification.

Many relations extend to wild ramification cases.

Group-theoretic calculations underpin the results.

Abstract

Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.