Finite radical field extensions and crossed homomorphisms
Let \(K^*\) denote the multiplicative group of nonzero elements of a field \(K\), and call \(K\) a radical extension of a subfield \(F\) if \(K=F(\Delta)\) for a subgroup \(\Delta\) of \(K^*\) such that \(\Delta\) contains \(F^*\) and \(\Delta/F^*\) is a torsion group. Now let \(E\) be a finite Galois extension of \(F\) with Galois group \(G\), and let \(T(E/F)\), respectively \(\mu (E)\), be the subgroup of all elements \(x\) of \(E^*\) such that \(x^n\in F^*\), respectively \(x^n=1\), for some positive integer \(n\). By expanding concepts from Kummer theory, the authors establish an isomorphism between \(T(E/F)/F^*\) and the group of crossed homomorphisms, i.e. one cocycles, of \(G\) into \(\mu(E)\). This isomorphism is used to characterize the subfields of \(E\) which are radical extensions of \(F\). If \(L\) is a finite field extension of \(F\) such that \(L\cup E=F\) and \(\mu(LE) =\mu(E)\), this characterization is employed to obtain a one-to-one correspondence between the subfields of \(E\) which are radical extensions of \(F\) and the subfields of \(LE\) which are radical extensions of \(L\). The authors obtain several more technical applications of this characterization.
scientific article; zbMATH DE number 955260