  Permutation trinomials over finite fields consititute an active research due
to their simple algebraic form, additional extraordinary properties and their
wide applications in many areas of science and engineering. In the present
paper, six new classes of permutation trinomials over finite fields of even
characteristic are constructed from six fractional polynomials. Further, three
classes of permutation trinomials over finite fields of characteristic three
are raised. Distinct from most of the known permutation trinomials which are
with fixed exponents, our results are some general classes of permutation
trinomials with one parameter in the exponents. Finally, we propose a few
conjectures.
