ABSTRACT
In this chapter we investigate finite Galois extensions which are radical, Kneser, or G-Cogalois, in terms of crossed homomorphisms. The results of this chapter are based on the description, provided in Section 6.1, of the Cogalois group https://www.w3.org/1998/Math/MathML"> C o g ( E / F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175695/6f0d74fd-b81c-4cd4-b8d9-805c7c0522ee/content/eq3730.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of any finite Galois extension E/F by means of crossed homomorphisms of the Galois group https://www.w3.org/1998/Math/MathML"> G a l ( E / F ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175695/6f0d74fd-b81c-4cd4-b8d9-805c7c0522ee/content/eq3731.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with coefficients in the group μ(E) of all roots of unity in E. This description, which is actually a reformulation of the Hilbert’s Theorem 90 in terms of Cogalois groups, states that there exists a canonical group isomorphism https://www.w3.org/1998/Math/MathML"> C o g ( E / F ) ≅ Z 1 ( G a l ( E / F ) , μ ( E ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175695/6f0d74fd-b81c-4cd4-b8d9-805c7c0522ee/content/eq3732.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
