  The effective field theory approach to high temperature field theory can be
used to study the phase transition in theories with spontaneously broken
symmetry. I construct a sequence of two effective three--dimensional field
theories which are valid on successively longer distance scales for a specific
model: $N$ charged scalars coupled to a U(1) gauge field. The resulting
effective Lagrangian can be used to investigate the phase transition, in
particular the order of the phase transition as a function of $N$, using
lattice simulations.
