Let \(L(y)=0\) be a homogeneous linear differential equation and \(R(z)=0\) be the associated Riccati equation. The paper deals with the problem of finding the possible degrees of the minimal polynomial of an algebraic solution of \(R(z)=0\). Supposing that \(L(y)\) is irreducible and \(L(y)=0\) has a Liouvillian solution, the author obtains sharp bounds for the above degrees and solves the problem completely when the order of \(L(y)\) is 3. This information may be useful for setting up an algorithm like \textit{J. Kovacic'} algorithm [J. Symb. Comput. 2, 3-43 (1986; Zbl 0603.68035)]. For references on this subject see \textit{M. F. Singer} [J. Symb. Comput. 11, No. 3, 251-273 (1991)].