Some results on radical extensions
Let \(F\) be a field, \(n\) a positive integer, and \(\zeta_ n\) a primitive \(n\)th root of unity. This paper is concerned with determining conditions on a field extension \(K/F\) which make it a radical extension. All field extensions are assumed to be separable. If \(K\) is a field, then \(K^*\) denotes the multiplicative group of nonzero elements of \(K\). If \(\Omega\) is a field, \(\mu(\Omega)\) denotes the group of roots of unity in \(\Omega\). Given a group \(G\) and a \(G\)-module \(A\), \(Z^ 1(G,A)\) denotes the group of 1-cocycles with coefficients in \(A\). The group of coboundaries is denoted by \(B^ 1(G,A)\) and the first cohomology group of \(G\) with coefficients in \(A\) is defined by \(H^ 1(G,A)= Z^ 1(G,A)/B^ 1(G,A)\). The author proves the following results. Let \(\Omega/F\) be a finite Galois extension and \(G=\text{Gal} (\Omega/F)\). Let \(H\leq G\) and \(K\) be the fixed field of \(H\). Then \(K/F\) is a radical extension if and only if there exists \(\chi\in Z^ 1(G, \mu(\Omega))\) with \(H= \text{Ker } \chi\), where \(\text{Ker } \chi= \{\sigma\in G\): \(\chi(\sigma)=1\}\). Let \(\Omega/F\) be a Galois extension and assume that there exists a finite extension \(L/F\) satisfying the properties that \(L\cap \Omega= F\) and \(\mu(L\Omega)= \mu(\Omega)\). Then there is a bijection between the radical extensions of \(F\) contained in \(\Omega\) and the radical extensions of \(L\) contained in \(L\Omega\). This bijection preserves the degree of the extensions. Let \(K/F\) be of degree \(n\) with \(\Omega\) the Galois closure over \(F\). Suppose that \(\text{char }F\) does not divide \(n\) and there exists a finite extension \(L/F\) such that \(\Omega(\zeta_ n)\cap L=F\) and \(LK= L(\alpha^{1/n})\) for some \(\alpha\in L\). Then \(K= F(a^{1/n})\) for some \(a\in F\). Let \(K/F\) be an extension of degree \(n\). Assume that \(K\cap F(\zeta_ n) =F\) and \(K(\zeta_ n)/ F(\zeta_ n)\) is cyclic. Then \(K(\zeta_ n)/F\) is normal and moreover, \(K(\zeta_ n)/F\) is abelian if and only if \(K/F\) is cyclic. Also \(K/F\) is radical if and only if \(b_ \sigma\equiv 1\pmod n\) for all \(\sigma\in \text{Gal}(F(\zeta_ n) /F)\).
scientific article; zbMATH DE number 554223