ABSTRACT
The prototype of an infinite G-Cogalois extension is, by Kummer Theory, any infinite classical Kummer extension. In this section we show that the essential part of the Infinite Kummer Theory can be immediately deduced from our Cogalois Theory using the Infinite n-Purity Criterion given in Section 12.1. Moreover, this criterion allows us to provide large classes of infinite G-Cogalois extensions which generalize or are closely related to classical infinite Kummer extensions: infinite generalized Kummer extensions, infinite Kummer extensions with few roots of unity, and infinite quasi-Kummer extensions. The prototype of an infinite Kummer extensions with few roots of unity is any subextension of R/Q of the form https://www.w3.org/1998/Math/MathML"> Q a i n ∣ i ∈ I / Q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175695/6f0d74fd-b81c-4cd4-b8d9-805c7c0522ee/content/eq6588.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> a i ∣ i ∈ I https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175695/6f0d74fd-b81c-4cd4-b8d9-805c7c0522ee/content/eq6589.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is an arbitrary nonempty set of strictly positive rational numbers. Notice that, in general, these extensions are not Galois if n ⩾ 3.
