  We investigate the phase structure of SU(3) gauge theory in four and five
dimensions with one compact dimension by using perturbative one-loop and
PNJL-model-based effective potentials, with emphasis on spontaneous gauge
symmetry breaking. When adjoint matter with the periodic boundary condition is
introduced, we have rich phase structure in the quark-mass and compact-size
space with gauge-symmetry-broken phases, called the $SU(2)\times U(1)$ split
and the $U(1)\times U(1)$ re-confined phases. Our result is qualitatively
consistent with the recent lattice calculations. When fundamental quarks are
introduced in addition to adjoint quarks, the split phase becomes more dominant
and larger as a result of explicit center symmetry breaking. We also show that
another $U(1)\times U(1)$ phase (pseudo-reconfined phase) with negative vacuum
expectation value of Polyakov loop exists in this case. We study chiral
properties in these theories and show that chiral condensate gradually
decreases and chiral symmetry is slowly restored as the size of the compact
dimension is decreased.
