Computation of unirational fields

TL;DR

This paper introduces an algorithm leveraging Groebner bases to compute all algebraic intermediate subfields in separably generated unirational field extensions, with potential extensions to finitely generated algebras.

Contribution

It presents a novel algorithm for computing algebraic intermediate subfields in unirational fields using Groebner bases, including primitive element computation and extension to K-algebras.

Findings

Algorithm successfully computes all algebraic intermediate subfields.

Method applies to separably generated unirational fields, including zero characteristic.

Extension of the method to finitely generated K-algebras is feasible.

Abstract

One of the main contributions which Volker Weispfenning made to mathematics is related to Groebner bases theory. In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.