  Let S be a subdivision of the plane into polygonal regions, where each region
has an associated positive weight. The weighted region shortest path problem is
to determine a shortest path in S between two points s, t in R^2, where the
distances are measured according to the weighted Euclidean metric-the length of
a path is defined to be the weighted sum of (Euclidean) lengths of the
sub-paths within each region. We show that this problem cannot be solved in the
Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one
can compute exactly any number that can be obtained from the rationals Q by
applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for
any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's
technique.
