  We study a notion of indecomposability in differential algebraic groups which
is inspired by both model theory and differential algebra. After establishing
some basic definitions and results, we prove an indecomposability theorem for
differential algebraic groups. The theorem establishes a sufficient criterion
for the subgroup of a differential algebraic group generated by an infinite
family of subvarieties to be a differential algebraic subgroup. This theorem is
used for various definability results. For instance, we show every
noncommutative almost simple differential algebraic group is perfect, solving a
problem of Cassidy and Singer. We also establish numerous bounds on Kolchin
polynomials, some of which seem to be of a nature not previously considered in
differential algebraic geometry; in particular, we establish bounds on the
Kolchin polynomial of the generators of the differential field of definition of
a differential algebraic variety.
