  The existence of a vertical age gradient in the Milky Way disc has been
indirectly known for long. Here, we measure it directly for the first time with
seismic ages, using red giants observed by Kepler. We use Stroemgren photometry
to gauge the selection function of asteroseismic targets, and derive colour and
magnitude limits where giants with measured oscillations are representative of
the underlying population in the field. Limits in the 2MASS system are also
derived. We lay out a method to assess and correct for target selection effects
independent of Galaxy models. We find that low mass, i.e. old red giants
dominate at increasing Galactic heights, whereas closer to the Galactic plane
they exhibit a wide range of ages and metallicities. Parametrizing this as a
vertical gradient returns approximately 4 Gyr/kpc for the disc we probe,
although with a large dispersion of ages at all heights. The ages of stars show
a smooth distribution over the last 10 Gyr, consistent with a mostly quiescent
evolution for the Milky Way disc since a redshift of about 2. We also find a
flat age-metallicity relation for disc stars. Finally, we show how to use
secondary clump stars to estimate the present-day intrinsic metallicity spread,
and suggest using their number count as a new proxy for tracing the ageing of
the disc. This work highlights the power of asteroseismology for Galactic
studies; however, we also emphasize the need for better constraints on stellar
mass-loss, which is a major source of systematic age uncertainties in red giant
stars.
