Radical extensions and crossed homomorphisms (Q2748271)

Previous results have shown a close connection between the group of one cocycles and radical extensions inside a Galois extension. In this paper, the authors present some results characterizing radical extensions inside a Galois extension using mainly the group of crossed homomorphisms. Let \(S/F\) denote the Galois extension with Galois group \(G\) and let \(\mu(S)\) denote the group of roots of unity in \(S\). The authors use the group \(Z^1(G,\mu(S))\) of crossed homomorphisms to study radical extensions inside \(S\). They also characterize cubic radical extensions and provide an example to show that the result can not be extended for higher degree extensions.