Abstract
Let G be a divisible ordered Abelian group, and let K be a field. The Hahn field K((G)) is a field of formal power series, with terms corresponding to elements in a well ordered subset of G and the coefficients coming from K. Ideas going back to Newton show that if K is either algebraically closed of characteristic 0, or real closed, then the same is true for K((G)). Results of Mourgues and Ressayre [11] led us to look for bounds on the lengths of roots of a polynomial, in terms of the lengths of the coefficients [5, 6]. In the present paper, we give an introduction to Hahn fields, we indicate how well quasi-orderings arise when we try to bound the lengths of sums and products, and we re-work, in a more general way, a technical theorem from [6] that gives information on the root-taking process.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Ash, C. J., & Knight, J. F. (2000). Computable structures and the hyperarithmetical hierarchy. Elsevier.
Carruth, P. (1942). Arithmetic of ordinals with application to the theory of ordered abelian groups. Bulletin of the American Mathematical Society, 48, 262–271.
deJongh, D., & Parikh, R. (1977). Well partial orderings and hierarchies. Proceedings Koninklijke Nederlandse Akademie Sci. Series A, 80, 195–207.
Hahn, H. (1995). Über die nichtarchimedischen Größensysteme. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse (Wien. Ber.) vol. 116(1907), pp. 601–655, reprinted in H. Hahn, Gesammelte Abhandlungen I. Springer.
Knight, J. F., & Lange, K. (2013). Complexity of structures associated with real closed fields. Proceedings of the London Mathematical Society, 107, 177–197.
Knight, J. F., & Lange, K. (2019). Lengths of developments in \(K((G))\). Selecta Mathematica, 25.
Knight, J. F., & Lange, K., Truncation-closed subfields of a Hahn field. pre-print.
Kříš, I., & Thomas, R., Ordinal types in Ramsey Theory and well-partial-ordering theory. In Mathematics of Ramsey Theory, ed. by J. Nešetřil & V. Rödl, pp. 57–95.
MacLane, S. (1939). The universality of formal power series fields. Bulletin of the American Mathematical Society, 45, 888–890.
Mal’tsev, A. L. (1948). On embedding of group algebras in a division algebra. (Russian) Doklady Akademii Nauk SSSR (N.S.), 60, 1499–1501.
Mourgues, M. H., & Ressayre, J.-P. (1993). Every real closed field has an integer part. Journal of Symbolic Logic, 58, 641–647.
Neumann, B. H. (1949). On ordered division rings. Transactions of the American Mathematical Society, 66, 202–252.
Newton, I. (1960). Letter to Oldenburg dated 1676 Oct 24 (pp. 126–127). The Correspondence of Isaac Newton II: Cambridge University Press.
Pohlers, W. (1980). Proof theory: An introduction. Berlin: Springer.
Puiseux, V. A. (1850). Recherches sur les fonctions algébriques. Journal de Mathématiques Pures et Appliquées, 15, 365–480.
Puiseux, V. A. (1851). Nouvelles recherches sur les fonctions algébriques. Journal de Mathématiques Pures et Appliquées, 16, 228–240.
Ressayre, J.-P. (1973). Boolean models and infinitary first order languages. Annals of Mathematical Logic, 6, 41–92.
Ressayre, J.-P. (1977). Models with compactness properties relative to an admissible language. Annals of Mathematical Logic, 11, 31–55.
Schmidt, D. (1978). Associative ordinal functions, well partial orderings, and a problem of Skolem. Z. Math. Logik Grundlagen Math., 24, 297–302.
Schmidt, D. (1979). Well partial orderings and their maximal order types. Heidelberg: Habilitationsschrift.
Schmidt, D. (1981). The relation between the height of a well-founded partial ordering and the order types of its chains and antichains. Journal of Combinatorial Theory, Series B, 31, 183–189.
Starchenko, S., Notes on roots of polynomials.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Knight, J.F., Lange, K. (2020). Well Quasi-orderings and Roots of Polynomials in a Hahn Field. In: Schuster, P., Seisenberger, M., Weiermann, A. (eds) Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-030-30229-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-30229-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30228-3
Online ISBN: 978-3-030-30229-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)