Irreducible Linear Differential Equations of Prime Order

Abstract

With the exception of a finite set of finite differential Galois groups, if an irreducible linear differential equation L(y) = 0 of prime order with unimodular differential 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 coefficients of L (y) = 0 are in Q (α)(x) ⊂ Q(x) this unique minimal polynomial is also defined over Q(α)(x). In the finite number of exceptions all solutions of L(y) = 0 are algebraic and in each case one can apriori give an extension Q(β)(x) over which the minimal polynomial of an algebraic solution of L(y) = 0 can be computed.