On computing a separating transcendence basis

Abstract

R a i n e r S t e i n w a n d t I n s t i t u t fiir A l g o r i t h m e n u n d K o g n i t i v e S y s t e m e , Prof. Dr. T h . B e t h , A r b e i t s g r u p p e C o m p u t e r~l g e b r a , F a k u l t~t fiir I n f o r m a t i k , Universit/it K a r l s r u h e , G e r m a n y AbstractLet k(x~,..., x,~)/k be a finitely generated field extension, and gl,... ,gr e k(g). For k ( x l , . . . ,x,~)/k(g:,... ,g~) being separably generated (which in particular includes the case char(k) = 0) we give a method to compute the transcendence degree and a separating transcendence basis of this extension by means of simple linear algebra techniques.

Computation of unirational fields

“…We can compute transcendence and algebraic degrees of unirational fields, decide whether an element is transcendental or algebraic over a field, compute its minimum polynomial in the latter case, and decide membership. Moreover, we can compute bases in the separable case without using, properly, Gröbner bases, see [Ste00]. The next step is solving the problem when the given extension is algebraic.…”

“…The results that we describe now will allow us to compute a separable basis and the transcendence degree of a separable extension without computing Gröbner bases, greatly increasing the efficiency of our computations. See (Weil, 1946) and (Steinwandt, 2000) for more details about these techniques.…”