SOME EXAMPLES IN COGALOIS THEORY WITH APPLICATIONS TO ELEMENTARY FIELD ARITHMETIC

Abstract

The aim of this paper is to provide some examples in Cogalois Theory showing that the property of a field extension to be radical (resp. Kneser, or Cogalois) is not transitive and is not inherited by subextensions. Our examples refer especially to extensions of type [Formula: see text]. We also effectively calculate the Cogalois groups of these extensions. A series of applications to elementary arithmetic of fields, like: • for what n, d ∈ ℕ* is [Formula: see text] a sum of radicals of positive rational numbers • when is [Formula: see text] a finite sum of monomials of form [Formula: see text], where r, j1,…, jr ∈ ℕ*, c ∈ ℚ*, and [Formula: see text] are also presented.