An algorithmic construction of cyclic p-extensions of fields,with characteristic different from p,not containing the pth roots of unity

摘要

Since the nineteenth century, when Kummer theory was first developed,we know how to build the cyclic p-extensions of fields E/F containing suffi-ciently many roots of unity, more precisely when F contains the p~n th rootswhere p~n= E : F]is the degree ofE/F(cf. [6, p. 289]). In 1989, Karpilovsky[5, p. 389] set the problem of finding an explicit description of all cyclic p-extensions. In 2002, the author [7] gave an algorithmic construction of anycyclic p-extension of fields with characteristic different from p, containingonly the pth roots of unity. The next and final step is to eliminate any prim-itive pth root of unity in the extension. This is what is done in the theoremstated below. The method uses the notion of a Galois average introducedin [8] (see also [4]). As a corollary for p = 3, we exhibit an algorithmiccomputable primitive element for any cyclic 3-extension.