Vol. 63, No. 2, 1976

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The existence of natural field structures for finite dimensional vector spaces over local fields

Mitchell Herbert Taibleson

Vol. 63 (1976), No. 2, 545–551

Let K be a local field (e.g., a p-adic or p-series field) and n a positive integer. Let Kbe the unique (up to isomorphism) unramified extension of K. It is shown that the natural (modular) norm of Kis the n-th power of the usual (l) vector space norm of Kwhen Kis viewed as an n-dimensional vector space over K. Further, the two distinct descriptions of the dual of K(which is isomorphic to K) that arise from the field model and vector space model are isomorphic under a K-linear isomorphism of Kas a vector space over K, and the isomorphism is norm preserving.

Mathematical Subject Classification 2000
Primary: 12B40, 12B40
Secondary: 43A70
Received: 7 August 1975
Published: 1 April 1976
Mitchell Herbert Taibleson