Exponentiation in power series fields

Kuhlmann, Franz-Viktor; Kuhlmann, Salma; Shelah, Saharon

doi:10.1090/S0002-9939-97-03964-6

Exponentiation in power series fields
HTML articles powered by AMS MathViewer

by Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon Shelah

Proc. Amer. Math. Soc. 125 (1997), 3177-3183

DOI: https://doi.org/10.1090/S0002-9939-97-03964-6

PDF | Request permission

Abstract:

We prove that for no nontrivial ordered abelian group $G$ does the ordered power series field $\mathbb {R}((G))$ admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field $k$, no exponential on $k((G))$ is compatible, that is, induces an exponential on $k$ through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

References

Norman L. Alling, On exponentially closed fields, Proc. Amer. Math. Soc. 13 (1962), 706–711. MR 141661, DOI 10.1090/S0002-9939-1962-0141661-0
Isidore Fleischer, Maximality and ultracompleteness in normed modules, Proc. Amer. Math. Soc. 9 (1958), 151–157. MR 93693, DOI 10.1090/S0002-9939-1958-0093693-8
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
Harry Gonshor, An introduction to the theory of surreal numbers, London Mathematical Society Lecture Note Series, vol. 110, Cambridge University Press, Cambridge, 1986. MR 872856, DOI 10.1017/CBO9780511629143
Hahn, H. Über die nichtarchimedischen Größensysteme, S.-B. Akad. Wiss. Wien, math.-naturw. Kl. Abt. IIa, 116 (1907), 601–655.
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Manfred Knebusch and Michael J. Wright, Bewertungen mit reeller Henselisierung, J. Reine Angew. Math. 286(287) (1976), 314–321 (German). MR 419419
Kuhlmann, S. On the structure of nonarchimedean exponential fields I, Archive for Math. Logic 34 (1995), 145–182.
Kuhlmann, F.-V. – Kuhlmann, S. On the structure of nonarchimedean exponential fields II, Comm. in Algebra 22(12) (1994), 5079–5103.
Kuhlmann, F.-V. – Kuhlmann, S. The exponential rank of nonarchimedean exponential fields, preprint.
Kuhlmann, F.-V. – Kuhlmann, S. – Shelah, S. Functional equations for lexicographic products, preprint.
Detlef Laugwitz, Eine nichtarchimedische Erweiterung angeordneter Körper, Math. Nachr. 37 (1968), 225–236 (German). MR 242798, DOI 10.1002/mana.19680370308
Laugwitz, D. Tullio Levi-Civita’s work on nonarchimedean structures, in: Tullio Levi-Civita, Convegno internazionale celebrativo del centenario della nascita, Academia Nazionale dei Lincei, Atti dei Convegni Lincei 8 (1973), 297–312.
Levi-Civita, T. Sugli infiniti ed infinitesimi attuali quali elementi analitici (1892-1893), opere mathematiche, vol. 1, Bologna (1954), 1–39.
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Abraham Robinson, Function theory on some nonarchimedean fields, Amer. Math. Monthly 80 (1973), no. 6, 87–109. Papers in the foundations of mathematics. MR 330126, DOI 10.2307/3038223