Gröbner Bases Applied to Finitely Generated Field Extensions

@article{MllerQuade2000GrbnerBA,
  title={Gr{\"o}bner Bases Applied to Finitely Generated Field Extensions},
  author={J{\"o}rn M{\"u}ller-Quade and Rainer Steinwandt},
  journal={J. Symb. Comput.},
  year={2000},
  volume={30},
  pages={469-490}
}
Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (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… 
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