  We evaluate exactly both the non-relativistic and relativistic fermion
determinant in 2+1 dimensions in a constant background field at finite
temperature. The effect of finite chemical potential is also considered. In
both cases, the systems are decoupled into an infinite number of 1+1 fermions
by Fourier transformation in the $\beta$-variable. The total effective actions
demonstrate non-extensiveness in the $\beta$ dimension.
