2025/11/04 by Perucca, Antonella
Mathematics · #Finite Group Theory Research #Algebraic Geometry and Number Theory #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2511.02498
Let K be a field, fix an algebraic closure K, and let G be a subgroup of K^×. We are able to give a closed formula for the ratio between the degree [K(G):K] and the index |GK^×:K^×|, 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^×/K^×) 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.