  A derivation of the ionization rate for the hydrogen-like ion in the strong
linearly polarized laser field is presented. This derivation utilizes the
famous Keldysh probability amplitude in the length gauge (in the dipole
approximation) and without Coulomb effects in the final state of the ionized
electron. No further approximations are being made, because the amplitude has
been expanded in the double Fourier series in a time domain (with the help of
the generalized Bessel functions). Thus, our theory has no other limitations
characteristic of the original Keldysh theory. We compare our "exact" theory
with the original Keldysh one, studying photoionization energy spectra and
total ionization rates. We show breakdown of the original Keldysh theory for
higher frequencies. In the barrier-suppresion regime the "exact" Keldysh theory
gives results closer to well-known numerical or other analytical results.
