Some Examples in Cogalois Theory With Applications To Elementary Fleld 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 Q(root r + root d)/Q. We also effectively calculate the Cogalois groups of these extensions. A series of applications to elementary arithmetic of fields, like: for what n, d is an element of N* is root n + root d a sum of radicals of positive rational numbers when is (n0)root a(0) a finite sum of monomials of form c center dot(n1)root a(1)(j1) ... (nr)root a(r)(jr), where r, j(1), ... , j(r) is an element of N*, c is an element of Q*, and a(0), ... , a(r) is an element of Q(+)(*) are also presented.