![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
Abstract
This paper investigates the converse to a theorem which states that a differential domain D finitely generated over a field F of characteristic 0 with no nonzero proper differential ideals will have constants of differentiation C only in the field in question. The converse is known to be false but the question of whether the differential domain D can be finitely extended within its quotient field E to a domain with no nonzero proper differential ideals was raised by Magid (Magid, A. (1991). Lectures on Differential Galois Theory. University Lecture Series, Vol. 7, Providence: AMS). Two examples are given with different properties to show that the answer is no. It is shown to be true under hypotheses which include restrictions on F and limiting the number of generators of D over C to two or the transcendence degree of D over C to one.