IRREDUCIBLE LINEAR-DIFFERENTIAL EQUATIONS OF PRIME-ORDER

With the exception of a finite set of finite differential Galois group s, if an irreducible linear differential equation L(y) = 0 of prime or der with unimodular differential Galois group has a Liouvillian soluti on, then all algebraic solutions of smallest degree of the associated Riccati equation are solutions of a unique minimal polynomial. If the coefficients of L(y) = 0 are in Q(alpha)(x) subset of (Q) over bar(x) this unique minimal polynomial is also defined over Q(alpha)(x). In th e finite number of exceptions all solutions of L(y) = 0 are algebraic and in each case one can apriori give an extension Q(beta)(x) over whi ch the minimal polynomial of an algebraic solution of L(y) = 0 can be computed.