Gröbner Bases Applied to Finitely Generated Field Extensions

Let k(~x) := k(x1, . . . , xn) be a finitely generated extension field of some field k, and denote by k(~g) := k(g1, . . . , gr) an intermediate field of k(~x)/k generated over k by some elements g1, . . . , gr ∈ k(~x). So geometrically, we may take ~g for rational functions on the variety determined by the generic point (~x). To determine whether the extension k(~x)/k(~g) is transcendental or algebraic and compute the transcendental/algebraic degree of this extension one can use Gröbner basis techniques involving so-called tag variables (see Kemper, 1993; Sweedler, 1993). The same techniques can also be applied to decide whether an element f ∈ k(~x) is algebraic over k(~g), and in the affirmative to find its minimal polynomial, for instance. For k(~x)/k being purely transcendental with transcendence basis {~x}, an alternate solution of these problems has been suggested in Müller-Quade and Steinwandt (1999). This approach is also based on Gröbner basis techniques, but in contrast to Sweedler (1993) and Kemper (1993) does not use tag variables. Due to the sensitivity of Buchberger’s algorithm to the number of variables involved, a generalization of these techniques to the case of not necessarily purely transcendental extensions k(~x)/k is desirable. Some results in this direction have been given in Müller-Quade et al. (1998). The aim of the present paper is to show that in fact most of the algorithms and results in Müller-Quade and Steinwandt (1999) can be extended to the situation where k(~x)/k is not necessarily purely transcendental. The key to the algorithms discussed is a correspondence between fields and certain ideals in polynomial rings. This correspondence can be made constructive by the use of Gröbner basis techniques. To derive the main results we use an approach which differs from the one in MüllerQuade and Steinwandt (1999) and Müller-Quade et al. (1998), as this allows less technical