Gröbner bases applied to finitely generated field extensions
In this article, the authors address practical problems in the computation with finitely generated field extensions. In particular, they give efficient algorithms to compute the (transcendence) degree of such an extensions, decide separability in positive characteristic, decide normality, give a field-membership test and present methods allowing the computation of minimal polynomials over subfields. The main technique used is that of Gröbner basis computations. This paper continues previous work of the authors [J. Symb. Comput. 27, 143-170 (1999; Zbl 0935.11046)]. The main difference between this approach and older ideas by \textit{M. E. Sweedler} [Lect. Notes Comput. Sci. 673, 66-75 (1993; Zbl 0813.13026)] and \textit{G. Kemper} [An algorithm to determine properties of field extensions lying over a ground field, IWR preprint 93-58, Heidelberg, Germany (1993)] is to avoid the use of tag-variables as they tend to slow down the necessary Gröber basis computations considerably. The article gives several worked out examples using Magma [\textit{W. Bosma, J. Cannon} and \textit{C. Playoust}, J. Symb. Comput. 24, 235-265 (1997; Zbl 0898.68039)] demonstrating the efficiency of their new methods, e.g. they verify the invariant field of a group of order \(248\) that was given in \textit{G. Kemper}'s PhD thesis [Noether's problem and generic polynomials (in German). Heidelberg (1994; Zbl 0830.12003)].
scientific article; zbMATH DE number 1538754