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
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by Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon Shelah PDF

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

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