Coefficient Fields of Solutions in Kovacicʹs Algorithm

In this paper we prove the following theorem: if the Riccati equation w′ + w2 = R(x), R ε Q(x), has algebraic solutions, then there exists a minimum polynomial defining such a solution whose coefficients lie at most in a cubic extension of the field Q. In Zharkov (1992), the same result was erroneously stated for, at most, quadratic extensions of Q. However, M. Singer discovered that in some cases the cubic extensions are necessary. Here we give a corrected and more detailed proof of the theorem.