Basic algorithms for rational function fields

Abstract

By means of Gröbner basis techniques algorithms for solving various problems concerning subfields double-struck K sign (g) := double-struck K sign (g1 , . . . , gm) of a rational function field double-struck K sign (x) := double-struck K sign (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 double-struck K sign (x)/double-struck K sign (g), deciding whether f ∈ double-struck K sign (x) is algebraic or transcendental over double-struck K sign (g), computing minimal polynomials, and deciding whether double-struck K sign (g) contains elements of a "particular structure", e.g. monic univariate 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 double-struck K sign (x) of this ideal and intermediate fields of double-struck K sign (g) and double-struck K sign (x) are given. In the last section some practical considerations concerning the use of the algorithms are discussed. © 1999 Academic Press.