Basic Algorithms for Rational Function Fields

By means of Gröbner basis techniques algorithms for solving various problems concerning subfields (g): = (g1, …,gm) of a rational function field (x): = (x1, …,xn) are derived: computing canonical generating sets, deciding field membership, computing the degree and separability degree resp. the transcendence degree and a transcendence basis of (x)/ (g), deciding whetherf (x) is algebraic or transcendental over (g), computing minimal polynomials, and deciding whether (g) contains elements of a “particular structure”, e.g. monicunivariate polynomials of fixed degree. The essential idea is to reduce these problems to questions concerning an ideal of a polynomial ring; connections between minimal primary decompositions over (x) of this ideal and intermediate fields of (g) and (x) are given. In the last section some practical considerations concerning the use of the algorithms are discussed.