Multiplicative groups of fields and hereditarily irreducible polynomials

TL;DR

This paper investigates the property of good heredity in fields using group theory, identifying which fields possess this property and providing counterexamples to common assumptions.

Contribution

It extends previous results on good heredity, classifies natural families of fields regarding this property, and constructs counterexamples to certain hypotheses.

Findings

Several natural families of fields are of good heredity.

Some fields do not have the good heredity property.

Counterexamples show certain intuitive expectations are false.

Abstract

In this paper we explore the concept of {\em good heredity} for fields from a group theoretic perspective. Extending results from \cite{alice}, we show that several natural families of fields are of good heredity, and some others are not. We also construct several examples to show that various wishful thinking expectations are not true.