The torsion group of a field defined by radicals
Let K/F be a finite, abelian extension of fields, and let p be a prime which divides the degree [K:F] of the extension. Let \(K^{\times}\) (resp. \(F^{\times})\) denote the multiplicative group of nonzero elements of K (resp. F), and let \(T_ p\) denote the p-torsion subgroup of the quotient group \(K^{\times}/F^{\times}\). If K does not contain a primitive p-th root of unity, then \(T_ p\) is trivial. If K, but not F, contains a primitive p-th root of unity, then \(T_ p\) is generated by the cosets of all \(p^ i-th\) roots of unity which are contained in K. Finally, if F contains a primitive p-th root of unity, then the authors obtain a complicated result which relates the rational decomposition of \(T_ p\) to the rational decomposition of the p-torsion subgroup of the Galois group of K/F. This information is used to determine the structure of the p-torsion subgroup of \(F(\alpha)^{\times}/F^{\times}\), where \(F(\alpha)\) is a separable, radical (i.e., \(\alpha^ m\in F\) for some positive integer m) extension of F.
scientific article; zbMATH DE number 3873456
scientific article