Linear relations among radicals

TL;DR

This paper provides a formula linking the degree of field extensions generated by radicals to the index of related multiplicative groups, clarifying all linear relations among radicals over a field.

Contribution

It introduces a closed formula for the ratio between extension degree and group index, explaining all linear relations among radicals beyond roots of unity.

Findings

Derived a closed formula for the degree-to-index ratio.

Identified all linear relations among radicals over a field.

Built on foundational work by Rybowicz, Kneser, and Schinzel.

Abstract

Let $K$ be a field, fix an algebraic closure $\overline{K}$, and let $G$ be a subgroup of $\overline{K}^\times$. We are able to give a closed formula for the ratio between the degree $[K(G):K]$ and the index $|GK^\times:K^\times|$, provided that the latter is finite. Our formula explains all the $K$-linear relations among radicals, which (beyond the ones stemming from the multiplicative group $GK^\times/K^\times$) are generated by relations among roots of unity and single radicals. Our work builds on results by Rybowicz, which in turn are based on work by Kneser and Schinzel.