TY - JOUR
T1 - Exponentiation in power series fields
AU - Kuhlmann, Franz Viktor
AU - Kuhlmann, Salma
AU - Shelah, Saharon
PY - 1997
Y1 - 1997
N2 - We prove that for no nontrivial ordered abelian group G does the ordered power series field 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.
AB - We prove that for no nontrivial ordered abelian group G does the ordered power series field 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.
KW - Convex valuations
KW - Lexicographic products
KW - Ordered exponential fields
KW - Power series fields
UR - http://www.scopus.com/inward/record.url?scp=21944432527&partnerID=8YFLogxK
U2 - 10.1090/s0002-9939-97-03964-6
DO - 10.1090/s0002-9939-97-03964-6
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:21944432527
SN - 0002-9939
VL - 125
SP - 3177
EP - 3183
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 11
ER -