  In this paper, extending our earlier program, we derive maximal canonical
extensions for multiplicative summations into algebraically closed fields. We
show that there is a well-defined analogue to minimal polynomials for a series
algebraic over a ring of series, the "scalar polynomial". When that ring is the
domain of a summation $\mathfrak{S}$, we derive the related concepts of the
$\mathfrak{S}$-minimal polynomial for a series, which is mapped by
$\mathfrak{S}$ to a scalar polynomial. When the scalar polynomial for a series
has the form $(t-a)^n$, $a$ is the unique value to which the series can be
mapped by an extension of the original summation.
