Liouvillian solutions of third order differential equations (Q1432884)

The author presents a refinement of some parts of the Kovacic algorithm to find Liouvillian solutions of third order equations. He proves that except for four finite groups and a reduction to the second order case, it is possible to deduce a formula in the imprimitive case. The author also provides some necessary conditions and simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups, by extracting ramification information from the character table. The author motivates his techniques by illustrative examples.