Linear relations among radicals

TL;DR

The paper addresses when $K$-linear relations among radicals are exclusively generated by multiplicative relations and provides a closed formula for the ratio $\frac{[K(G):K]}{|GK^{\times}:K^{\times}|}$ in terms of cyclotomic data and roots of unity. It develops a unified framework that handles odd and even $n$ separately, leverages Kneser–Kummer theory, and extends results of Rybowicz with Schinzel’s abelian radical extensions to the general case via a prime-power decomposition. The main contributions are (i) a precise, computable formula for the degree-index ratio that explains entanglement as relations among roots of unity and single radicals, (ii) a detailed treatment of the odd prime power and $2$-power cases, and (iii) a general theory for arbitrary $n$ showing divisibility properties and eventual growth behavior (MAMA-type results). This provides a comprehensive understanding of radical extensions, clarifying when nonmultiplicative $K$-linear relations arise and how roots of unity govern entanglement, with implications for the structure and growth of radical extensions over $K$.

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.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4: Kneser's Theorem
• Proposition 5
• proof
• Remark 6
• Theorem 7: Kummer theory
• proof
• Theorem 8: Kneser - Rybowicz