Formally Real Fields

Author

Gail L. Carns

Publication Date

8-9-1968

Abstract

This paper is concerned with the analysis of the set of orders on a formally real field using category and sheaf theory as tools. We topologize the set of orders in a way similar to that used in the prime spectrum of algebraic geometry and define a presheaf of groups using the notion of inverse limits. We then find necessary and sufficient conditions that these presheaves be sheaves. Next we notice that if F ⊆ Fⁱ respectively, fields, θ and θⁱ are the set of orders on F and Fⁱ respectively, and Γ and Γⁱ are the associate presheaves then there exists a natural continuous map ѱ: θⁱ→θ and a presheaf homomorphism from Γ to the direct image of Γⁱ under ѱ. The final section of the paper deals with locally good functions. This concept comes from the fact that for each x Є F there exists a function f_x:θ→R defined by f_x(α) = sup {r Є Q: x - r Є α}. For U open in θ a function f : U → R is defined to be locally good if for each α Є U there exists a neighborhood V of α and an x Є F such that f_|V = f_x|V and |f_x (β)| < for all β Є V: The set ẞ(U) for all locally good functions on U is a ring under pointwise operation and we obtain a sheaf ẞ on θ by assigning to each U open in θ the set ẞ(U) and for U ⊆ V with U and V open subsets of θ defining the map : ẞ(V) → ẞ(U) to be the restriction map. For F ⊆ Fⁱ and ẞ and ẞⁱ the associated sheaves there exists a sheaf homomorphism from ẞ to the direct image of ẞⁱ under ѱ. This produces a contravariant function A from the category of fields with inclusions to the category of ringed spaces and ringed space morphisms. Two orders α, β on F are defined to be equivalent, denoted α≈β, if for all x Є F we have that f_x(α) = f_x(β). The factor set of θ by ≈ is denoted θ/≈ and we define a sheaf of rings on θ/≈ in a way an analogous to the way ẞ was defined. If F ⊆ Fⁱ are fields, θ/≈ and θⁱ/≈ are the factor sets for the set of orders on F and Fⁱ respectively, and are the associated sheaves then we obtain a continuous map : θⁱ/≈ → θ/≈ and sheaf homomorphism from to the direct image of under thus obtaining another contravariant functor B from the category of fields with inclusions to the category of ringed spaces and ringed space morphisms. The canonical map from θ to θ/≈ is denoted by ϕ and for f Є (U) we find that f о ϕ Є ẞ(ϕ^-1(U)) and in this manner we define a sheaf homomorphism θ_F from to the direct image of ẞ under ϕ. Then {(ϕ: θ_F) : F is an orderable field} is a natural transformation from A to B. As an example we find that if F is the field of rational functions over the rational numbers then θ/≈ is homeomorphic to the one point compactification of the real numbers and f is an element of (U) if and only if f is a finite valued rational function on each connected component of U.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Donald Ward Dubois

Second Committee Member

Illegible

Third Committee Member

Illegible

Fourth Committee Member

Richard Clyde Metzler

Language

English

Document Type

Dissertation

Recommended Citation

Carns, Gail L.. "Formally Real Fields." (1968). https://digitalrepository.unm.edu/math_etds/226