  We show that an effective version of Siegel's Theorem on finiteness of
integer solutions and an application of elementary Galois theory are key
ingredients in a complexity classification of some Holant problems. These
Holant problems, denoted by Holant(f), are defined by a symmetric ternary
function f that is invariant under any permutation of the k >= 3 domain
elements. We prove that Holant(f) exhibits a complexity dichotomy. This
dichotomy holds even when restricted to planar graphs. A special case of this
result is that counting edge k-colorings is #P-hard over planar 3-regular
graphs for k >= 3. In fact, we prove that counting edge k-colorings is #P-hard
over planar r-regular graphs for all k >= r >= 3. The problem is
polynomial-time computable in all other parameter settings. The proof of the
dichotomy theorem for Holant(f) depends on the fact that a specific polynomial
p(x,y) has an explicitly listed finite set of integer solutions, and the
determination of the Galois groups of some specific polynomials. In the
process, we also encounter the Tutte polynomial, medial graphs, Eulerian
partitions, Puiseux series, and a certain lattice condition on the (logarithm
of) the roots of polynomials.
