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A Question About Differential Ideals

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.