We prove that if $G$ is a solvable group with rational characters and {\bf R} is a splitting field for $G$, then {\bf Q}$(2^{1/2})$ is also a splitting field for $G$ and we obtain some sufficient conditions which guarantee that an irréductible character $\Gamma$ of a group with rational characters has Schur indices $m_Q(\Gamma)=1$. These results are related to the Gow conjecture [2] wich asserts that for a solvable group whose characters are rational valued and {\bf R} is a splitting field for $G$, then {\bf Q} is also a splitting field for $G$.