the irrationality of sums of radicals via cogalois theory

28 Toma Albuwhere r, n 1 , . . . , n r ∈ N ∗ , a 1 , . . . , a r ∈ Q ∗ + and √ n i a i is the positive realn i -th root of a i for each i, 1 i r. For such an extension, the associatedgroup ∆ is the factor groupQ ∗ 〈 √ n 1a 1 , . . . , √ n ra r 〉/ Q ∗ .Roughly speaking, Cogalois Theory investigates finite radical extensions,i.e.,F ( √ n 1a 1 , . . . , √ n ra r )/Fwhere F is an arbitrary field, r, n 1 , . . . , n r ∈ N ∗ , a 1 , . . . , a r ∈ F ∗ and √ n i a i ∈Ω is an n i -th root of a i , ∀ i, 1 i r. In the most cases∆ = F ∗ 〈 √ n 1a 1 , . . . , nr√ a r 〉/F ∗ .In our opinion, this theory was born in 1986 when the fundamental paperof Cornelius Greither and David K. Harrison [15] has been published. Notethat, like in the case of Galois Theory, where an infinite Galois Theory exists,an infinite Cogalois Theory has been invented in 2001 by Albu and Ţena[5]. Further, the infinite Cogalois Theory has been generalized in 2005 toarbitrary profinite groups by Albu and Basarab [2], leading to a so calledabstract Cogalois Theory for such groups.We are now going to present the basic concept of Cogalois Theory, namelythat of G-Cogalois extension we referred after Remark 2.10. To do that,we need first to define the following notions: Cogalois group, radical extension,Cogalois extension, G-radical extension, G-Kneser extension, stronglyG-Kneser extension, and Kneser group.For any extension E/F we denoteT (E/F ) := { x ∈ E ∗ | x n ∈ F ∗ for some n ∈ N ∗ }.Clearly F ∗ T (E/F ) E ∗ , so it makes sense to consider the quotient groupT (E/F )/F ∗ , which is nothing else than the torsion group t(E ∗ /F ∗ ) of the