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, 888-890 (1939)] 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 γ is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than ω ωγ .