  We integrate out fast varying quantum fluctuations around static A_4 and A_i
fields for the SU(N) gauge group. By assuming that the gluon fields are slowly
varying but allowing for an arbitrary amplitude of A_4 we obtain two variants
of the effective high-temperature theory for the Polyakov line. One is the
effective action for the gauge-invariant eigenvalues of the Polyakov line, and
it is explicitly Z(N) symmetric. The other is the effective action for the
Polyakov line itself as an element of the SU(N). In this case the theory
necessarily includes the spatial components A_i to ensure its gauge invariance
under spatial gauge transformations. We derive the 1-loop effective action in
the `electric' and `magnetic' sectors, summing up all powers of A_4.
