Gröbner Bases Applied to Finitely Generated Field Extensions
✍ Scribed by Jörn Müller-Quade; Rainer Steinwandt
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 442 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0747-7171
No coin nor oath required. For personal study only.
✦ Synopsis
Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensions k( x)/k( g), resp. how to determine the (separable) degree if k( x)/k( g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k( x)/k( g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group.