Irreducible Linear Differential Equations of Prime Order

With the exception of a nite set of nite diierential Galois groups, if an irreducible linear diierential equation L(y) = 0 of prime order with unimodular diierential Galois group has a Liouvillian solution, then all algebraic solutions of smallest degree of the associated Riccati equation are solutions of a unique minimal polynomial. If the coeecients of L(y) = 0 are in Q()(x) Q(x) this unique minimal polynomial is also deened over Q()(x). In the nite number of exceptions all solutions of L(y) = 0 are algebraic and in each case one can apriori give an extension Q()(x) over which the minimal polynomial of an algebraic solution of L(y) = 0 can be computed.