Lengths of roots of polynomials in a Hahn field

Let $K$ be an algebraically closed field of characteristic $0$, and let $G$ be a divisible ordered Abelian group. Maclane [Bull. Am. Math. Soc., 45 (1939), 888—890] showed that the Hahn field $K((G))$ is algebraically closed. Our goal is to bound the lengths of roots of a polynomial $p(x)$ over $K((G))$ in terms of the lengths of its coefficients. The main result of the paper says that if $\gamma$ is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than $\omega^{\omega^\gamma}$.