  The classical Ritt's Theorems state several properties of univariate
polynomial decomposition. In this paper we present new counterexamples to
Ritt's first theorem, which states the equality of length of decomposition
chains of a polynomial, in the case of rational functions. Namely, we provide
an explicit example of a rational function with coefficients in Q and two
decompositions of different length.
  Another aspect is the use of some techniques that could allow for other
counterexamples, namely, relating groups and decompositions and using the fact
that the alternating group A_4 has two subgroup chains of different lengths;
and we provide more information about the generalizations of another property
of polynomial decomposition: the stability of the base field. We also present
an algorithm for computing the fixing group of a rational function providing
the complexity over Q.
