  The T-odd top-quark chromoelectric dipole moment (tCEDM) is probed through
top-quark-pair production via gluon fusion at the CERN Large Hadron Collider
(LHC) by considering the possibility of having polarized protons. The complete
analytic expressions for the tree-level helicity amplitudes of gg-> ttbar is
also presented. For the derived analytic results we determine the 1-sigma
statistical sensitivities to the tCEDM form factor for (i) typical CP-odd
observables composed of lepton and anti-lepton momenta from t and tbar
semileptonic decays for unpolarized protons, and (ii) a CP-odd event asymmetry
for polarized protons by using the so-called Berger-Qiu (BQ) parametrization of
polarized gluon distribution functions. We find that at the CERN LHC, the
CP-odd energy and angular correlations can put a limit of 10^{-18} to 10^{-17}
g_scm on the real and imaginary parts of the tCEDM, while the simple CP-odd
event asymmetry with polarized protons could put a very strong limit of
10^{-20} g_scm on the imaginary part of the tCEDM.
