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The Nakayama Map and Ramification for Maximally Complete Fields

Published online by Cambridge University Press:  20 November 2018

Murray A. Marshall
Murray A. Marshall*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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Let K be a maximally complete valued field and let L be a totally ramified Galois extension of K with Galois group G. Assume (i) the value group quotient of L|K is cyclic and (ii) there exists an unramified cyclic extension of K of the same degree as L. Then there is an isomorphism of Ga onto a subgroup A/N(L×) of K×/N(L×) which maps the ramification group Gi onto AiN(L×)/N(L×) for all i > 0 where Ai = {x ∊ A|v(x ‒ 1) ≧ i}. This generalizes certain results of Local Class Field Theory.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 917 - 919
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Geissinger, L., A reciprocity law for maximal fields, Trans. Amer. Math. Soc. 125 (1966), 422—431.Google Scholar
2. Marshall, M., Ramification theory for valuations of arbitrary rank, Can. J. Math. 26 (1974), 908–916.Google Scholar
3. Nakayama, T., Über die Beziehungen zwischen den Faktorensystemen und der Normklassengruppe eines galoisschen Erweiterungskörpers, Math. Ann. 112 (1936), 85–91.Google Scholar
4. Shilling, O. F. G., The theory of valuations. Math. Surveys, No. IV (Amer. Math. Soc. publ., Providence, 1950).Google Scholar