Journal of Symbolic Logic

Elementary Properties of Power Series Fields over Finite Fields

Franz-Viktor Kuhlmann

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In spite of the analogies between $\mathbb{Q}_p$ and $\mathbb{F}_p ((t))$ which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for $\mathbb{Q}_p$ to the case of $\mathbb{F}_p((t))$ does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on $\mathbb{F}_p((t))$. We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic.

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J. Symbolic Logic, Volume 66, Issue 2 (2001), 771-791.

First available in Project Euclid: 6 July 2007

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Kuhlmann, Franz-Viktor. Elementary Properties of Power Series Fields over Finite Fields. J. Symbolic Logic 66 (2001), no. 2, 771--791.

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