  We discuss a natural scenario to solve the strong CP problem in the framework
of the higher dimensional gauge theory. An axion-like field $A_y$ has been
built-in as the extra-space component of the higher dimensional gauge field.
The coupling of $A_y$ with gluons is attributed to the radiatively induced
"Chern-Simons" (CS) term. We adopt a toy model with some unknown gauge symmetry
U(1)$_X$. The CS term is obtained in two ways: first by a concrete 1-loop
calculation and next by use of the Fujikawa's method to deal with the chiral
anomaly in 4D space-time. The obtained results are identical, which implies
that the radiative correction to the CS term is "1-loop exact" and is also free
from UV-divergence even though the theory itself is non-renormalizable. As a
novel feature of this scenario, such obtained CS term is no longer linear in
the field $A_y$ as in the usually discussed CS term in 5D space-time but a
periodic function of $A_y$, since $A_y$ has a physical meaning as the
Wilson-loop phase. We argue how such novel feature of this scenario causes the
modification of the ordinary solutions of the strong CP problem based on the
axion fields.
