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 )...

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Bibliographic Details
Published in:Algebra and logic 2021-05, Vol.60 (2), p.95-107
Main Authors: Knight, J. F., Lange, K.
Format: Article
Language:English
Subjects:
Abelian groups
Algebra
Fields, Algebraic
Group theory
Mathematical analysis
Mathematical Logic and Foundations
Mathematical research
Mathematics
Mathematics and Statistics
Polynomials
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Summary: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 .
ISSN:0002-5232
1573-8302
DOI:10.1007/s10469-021-09632-0