Geometric theory of valued fields

Nianzheng Liu, Purdue University

Abstract

This thesis gives an account of algebraic and analytic results in the geometric theory of valued fields, in an approach that uses elementary arguments, and explicit procedures as a unifying principle. Instead of working in general valued fields, I state the results in terms of the fields of p-adic numbers $\doubq\sb{p}$ and the $\doubc\sb{p}$, the completion of the algebraic closure of the field of the p-adic numbers. As many classical notions have their p-adic counterparts, we present in the first part a p-adic analog of a separation theorem and a stratifying procedure for p-adic semialgebraic sets. In the second part of my thesis, I studied the theory of p-adic semi-analytic sets. I obtained two main results of the theory of p-adic semi-analytic sets: p-adic semi-analytic sets may be stratified into semi-analytic subsets with the "condition of the frontier", and the closure of a p-adic semi-analytic set is just a p-adic semi-analytic set. The last part of the thesis is concerned with geometry over $\doubc\sb{p}$--rigid semi- and subanalytic geometry. The following results are included: Several rigid analogs of semi-analytic theorems, a proof of an analog in the rigid case of Milnor's curve selection lemma and a Lojasiewicz inequality for rigid subanalytic sets.

Degree

Ph.D.

Advisors

Lipshitz, Purdue University.

Subject Area

Mathematics

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