Deciding Linear Disjointness of Finitely Generated Fields
- Jorn Muller-Quade ,
- Martin Roetteler
Proceedings of ISSAC'98 (Rostock) |
Published by ACM Press
The behaviour of two field extensions with respect to each other can be described by the notions of linear disjointness and freeness. This paper gives methods for effectively deciding linear disjointness and freeness for fields lying under a finitely generated field k(X)=Quot(k[X1, …, Xn]/I(X)).
Furthermore the methods developed can be used to decide for two fields if there exists a field over which they are linear disjoint. This field (if it exists) is always the intersection of the two fields given. Thus we are able to compute the intersection of finitely generated fields in this situation.
All methods used rely on a correspondence from (pairs of) fields to ideals namely the ideal of syzygies of the generators of one field which have coefficients lying in the other field. We thereby generalize existing correspondences associating single fields or field extensions to ideals. Our contribution concludes with an outlook to the problem of computing the intersection of fields in more general situations.