A proof by P. C. Rosenbloom of the fundamental theorem of algebra is
used in the paper for obtaining a subrecursive refinement of the
theorem. Arbitrary complex numbers are supposed to be represented as
limits of such sequences of rational ones that the distance between the
n-th term of the sequence and its limit is not greater than
the
reciprocal of n+1. In the case when polynomials of a fixed
degree are
considered, we show the existence of computable operators that belong
to the second Grzegorczyk class and transform any representations of
the coefficients of any monic polynomial into representations of its
roots.