Grobner bases applied to finitely generated field extensions

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) over right arrow/k (g) over right arrow, resp. how to determine the (separable) degree if k (x) over right arrow/k ( g) over right arrow is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field member ship. Finally, a connection between certain intermediate fields of k (x) ov er right arrow/k (g) over right arrow and a minimal primary decomposition o f a suitable ideal is described. For Galois extensions the field-ideal corr espondence can also be used to determine the elements of the Galois group. (C) 2000 Academic Press.