Valuations in algebraic field extensions

Extensions of valuations

Continuous valuations have been proposed by several authors as a way of modeling probabilistic non-determinism in programming language semantics. Let (X;O) be a topological space. A quasi-simple valuation on X is the sup of a directed family of simple valuations. We show that quasisimple valuations are exactly those valuations that extend to continuous valuations to the Alexandroff topology on ...

Geometric Constructions and Algebraic Field Extensions

In this paper, we study field extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions exist to three ancient Greek construction problems: squaring the circle, doubling the cube, and trisecting an angle.

Reverse Mathematics and Algebraic Field Extensions

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section §2, we show that WKL0 is equivalent to the ability to extend F -automorphisms of field extensions to automorphisms of F , the algebraic closure of F . Section §3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section §4, and the Gal...

INTERSECTIONS OF LOCAL ALGEBRAIC EXTENSIONS OF A HILBERTIAN FIELD* by

Parametrization of Algebraic Curves over Optimal Field Extensions

In this paper we investigate the problem of determining rational parametrizations of plane algebraic curves over an algebraic extension of least degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of definition on algebraic curves of genus 0. Consequently we are also able to decide parametrizability over the reals. We g...