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.

中文翻译：

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哈恩域中多项式根的长度

令K为特征为 0 的代数闭域，令G为可整除的有序阿贝尔群。麦克莱恩[公牛。是。数学。Soc., 45, 888-890 (1939)] 表明哈恩域K (( G )) 是代数闭的。我们的目标是根据系数的长度来限制K (( G )) 上多项式p ( x )的根长度。论文的主要结果是，如果𝛾是一个大于所有系数长度的极限序数，那么根的长度都小于ω ^ω𝛾。