  We use the Gaia data release 1 (DR1) to study the proper motion (PM) fields
of the Large and Small Magellanic Clouds (LMC, SMC). This uses the Tycho-Gaia
Astrometric Solution (TGAS) PMs for 29 Hipparcos stars in the LMC and 8 in the
SMC. The LMC PM in the West and North directions is inferred to be
$(\mu_W,\mu_N) = (-1.872 \pm 0.045, 0.224 \pm 0.054)$ mas/yr, and the SMC PM
$(\mu_W,\mu_N) = (-0.874 \pm 0.066, -1.229 \pm 0.047)$ mas/yr. These results
have similar accuracy and agree to within the uncertainties with existing
Hubble Space Telescope (HST) PM measurements. Since TGAS uses different methods
with different systematics, this provides an external validation of both data
sets and their underlying approaches. Residual DR1 systematics may affect the
TGAS results, but the HST agreement implies this must be below the random
errors. Also in agreement with prior HST studies, the TGAS LMC PM field clearly
shows the clockwise rotation of the disk, even though it takes the LMC disk in
excess of $10^8$ years to complete one revolution. The implied rotation curve
amplitude for young LMC stars is consistent with that inferred from
line-of-sight (LOS) velocity measurements. Comparison of the PM and LOS
rotation curves implies a kinematic LMC distance modulus $m-M = 18.54 \pm
0.39$, consistent but not yet competitive with photometric methods. These first
results from Gaia on the topic of Local Group dynamics provide an indication of
how its future data releases will revolutionize this field.
