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Basic Algorithms for Rational Function Fields

✍ Scribed by J. Müller-Quade; R. Steinwandt

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
411 KB
Volume
27
Category
Article
ISSN
0747-7171

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✦ Synopsis

By means of Gröbner basis techniques algorithms for solving various problems concerning subfields K (g) := K (g 1 , . . . , gm) of a rational function field K (x) := K (x 1 , . . . , 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 K (x)/K (g), deciding whether f ∈ K (x) is algebraic or transcendental over K (g), computing minimal polynomials, and deciding whether K (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 K (x) of this ideal and intermediate fields of K (g) and K (x) are given. In the last section some practical considerations concerning the use of the algorithms are discussed.

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