  We consider N=2 supersymmetric gauge theories perturbed by tree level
superpotential terms near isolated singular points in the Coulomb moduli space.
We identify the Seiberg-Witten curve at these points with polynomial equations
used to construct what Grothendieck called "dessins d'enfants" or "children's
drawings" on the Riemann sphere. From a mathematical point of view, the dessins
are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully
on them. We argue that the relation between the dessins and Seiberg-Witten
theory is useful because gauge theory criteria used to distinguish branches of
N=1 vacua can lead to mathematical invariants that help to distinguish dessins
belonging to different Galois orbits. For instance, we show that the
confinement index defined in hep-th/0301006 is a Galois invariant. We further
make some conjectures on the relation between Grothendieck's programme of
classifying dessins into Galois orbits and the physics problem of classifying
phases of N=1 gauge theories.
