Journal of Symbolic Logic

Elementary Properties of Power Series Fields over Finite Fields

Franz-Viktor Kuhlmann

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183746472

Mathematical Reviews number (MathSciNet)
MR1833477

Zentralblatt MATH identifier
0992.03046

JSTOR
links.jstor.org

Citation

Kuhlmann, Franz-Viktor. Elementary Properties of Power Series Fields over Finite Fields. J. Symbolic Logic 66 (2001), no. 2, 771--791. https://projecteuclid.org/euclid.jsl/1183746472


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