  The paper investigates the converse to the following theorem. Let R be a
differential domain R which is finitely generated over a differential field F
whose field of constants is algebraically closed of characteristic 0. If R has
no proper nonzero differential ideals, then the quotient field, E, of R has no
new constants. The converse is false, but a question was raised about the
existence of a finitely generated extension of R within E which has no proper
nonzero differential ideals when E has no new constants. This posted paper
gives one example to show this is false and two limited positive results in
Krull dimension one or two.
