Basic algorithms for rational function fields

By means of Grobner basis techniques algorithms for solving various problem s concerning subfields K(g) := K(g(1),...,g(m)) of a rational function fiel d K(x) := K(x(1),...,x(n)) are derived: computing canonical generating sets , deciding field membership, computing the degree and separability degree r esp. the transcendence degree and a transcendence basis of K(x)/K(g), decid ing whether f is an element of K(x) is algebraic or transcendental over K(g ), computing minimal polynomials, and deciding whether K(g) contains elemen ts of a "particular structure", e.g. monic univariate polynomials of fixed degree. The essential idea is to reduce these problems to questions concern ing an ideal of a polynomial ring; connections between minimal primary deco mpositions over K(x) of this ideal and intermediate fields of K(g) and K(x) are given. In the last section some practical considerations concerning th e use of the algorithms are discussed. (C) 1999 Academic Press.