The linear relations among radicals

Given a base field $K$, we consider its radicals, namely the elements $\alpha$ (in a given algebraic closure of $K$) such that $\alpha^n\in K$ for some integer $n$ coprime to the characteristic of $K$. The most classical case is covered by Kummer theory, which additionally requires that the $n$-th roots of unity are contained in $K$. We study the entanglement of radicals, namely their $K$-linear relations that do not stem from multiplicative relations. Building on Kneser's theorem on the linear independence of radicals and on work by Ribowicz, we are able to completely describe the entanglement of radicals.