  We investigate the integer Hall conductivity and longitudinal conductivity of
silicene under the magnetic field, electric field, and exchange field in this
letter. We focus not only on the low-temperature and $\delta$-function
impurities (i.e., independent of the scattering momentum) case, which only
exist the intra-Landau level transition, but also on the case of inter-Landau
level transition which also with the non-elastic scattering. The resulting
longitudinal conductivity is very different with the intra-Landau level one at
low-temperature. The exprssions of the Hall conductivity, longitudinal
conductivity, valley contributed Hall conductivity, and the spin or valley Hall
conductivity, are deduced in this letter. We also compute the dynamical
polarization under the magnetic field which is a important quantity and has
many exciting and novel properties, and with the screened scattering due to the
charged impurities. The polarization function here is also related to the
Landau level index under the magnetic field, and shows step-like feature rather
than the logarithmically divergenas which appears for the zero-magnetic field
case, and it's also naturally related to the conductivity in silicene. The
generalized Laguerre polynomial is used in both the longitudinal conductivity
and dynamical polarization function under magnetic field.
