In an effort to further understand the structure of effective actions for
fermions in an external gauge background at finite temperature, we study the
example of 1+1 dimensional fermions interacting with an arbitrary Abelian gauge
field. We evaluate the effective action exactly at finite temperature. This
effective action is non-analytic as is expected at finite temperature. However,
contrary to the structure at zero temperature and contrary to naive
expectations, the effective action at finite temperature has interactions to
all (even) orders (which, however, do not lead to any quantum corrections). The
covariant structure thus obtained may prove useful in studying 2+1 dimensional
models in arbitrary backgrounds. We also comment briefly on the solubility of
various 1+1 dimensional models at finite temperature.Comment: A few clarifying remarks added;21 page
