  Kepler ultra-high precision photometry of long and continuous observations
provides a unique dataset in which surface rotation and variability can be
studied for thousands of stars. Because many of these old field stars also have
independently measured asteroseismic ages, measurements of rotation and
activity are particularly interesting in the context of age-rotation-activity
relations. In particular, age-rotation relations generally lack good
calibrators at old ages, a problem that this Kepler sample of old-field stars
is uniquely suited to address. We study the surface rotation and photometric
magnetic activity of a subset of 540 solar-like stars on the main- sequence and
the subgiant branch for which stellar pulsations have been measured. The
rotation period was determined by comparing the results from two different
analysis methods: i) the projection onto the frequency domain of the
time-period analysis, and ii) the autocorrelation function (ACF) of the light
curves. Reliable surface rotation rates were then extracted by comparing the
results from two different sets of calibrated data and from the two
complementary analyses. We report rotation periods for 310 out of 540 targets
(excluding known binaries and candidate planet-host stars); our measurements
span a range of 1 to 100 days. The photometric magnetic activity levels of
these stars were computed, and for 61.5% of the dwarfs, this level is similar
to the range, from minimum to maximum, of the solar magnetic activity. We
demonstrate that hot dwarfs, cool dwarfs, and subgiants have very different
rotation-age relationships, highlighting the importance of separating out
distinct populations when interpreting stellar rotation periods. Our sample of
cool dwarf stars with age and metallicity data of the highest quality is
consistent with gyrochronology relations reported in the literature.
