  We compute the R\'enyi entropies of the massless Dirac field on the Euclidean
torus (the Lorentzian cylinder at non-zero temperature) for arbitrary spatial
regions. We do it by the resolvent method, i.e., we express the entropies in
terms of the resolvent of a certain operator and then use the explicit form of
that resolvent, which was obtained recently. Our results are different in
appearance from those already existing in the literature (obtained via the
replica trick), but they agree perfectly, as we show numerically for
non-integer order and analytically for integer order. We also compute the
R\'enyi mutual information, and find that, for appropriate choices of the
parameters, it is non-positive and non-monotonic. This behavior is expected,
but it cannot be seen with the simplest known R\'enyi entropies in quantum
field theory because they are proportional to the entanglement entropy.
