  Classifying the phases of gauge theories is hindered by the lack of local
order parameters. In particular, the standard Wilson's and 't Hooft's non-local
order parameters are known to be insufficient to explain the existence of the
plethora of phases that are found in supersymmetric gauge theories. Motivated
by these observations, we reanalyze the concept of gauge symmetry breaking
using Galois theory. Unlike the ordinary classical notion of unbroken gauge
group, the Galois symmetry makes sense in the full quantum theory and must be a
phase invariant. The algebraic structure underlying the space of vacua of
supersymmetric gauge theories, that we have developed recently, is precisely
designed to allow a rigorous mathematical implementation of these ideas.
