VECTOR-SPACES OVER UNSPECIFIED FIELDS

DAVID PIERCE

In the Elements, Euclid of Alexandria gives a geometric formulation of certain field-theoretic identities; for example, the identity

x^{2} − y^{2} = (x + y)(x − y)

is Euclid’s Proposition II.5, but Euclid expresses in terms of the squares and rectangles bounded by certain lines. At the beginning of the Geometry, René Descartes shows how lines can be multiplied to produce lines rather than rectangles, if one line is chosen as unit. The construction justifies Descartes’s algebraic treatment of geometry.

A model-theoretic version of Descartes’s construction is that the theory of vector-spaces (as one-sorted structures, but over unspecified fields) can be axiomatized in the signature of abelian groups with a predicate for parallelism (binary linear dependence). The existentially closed models of this theory are two-dimensional spaces (over algebraically closed fields). If a predicate for n-ary linear dependence is introduced to the language, then the existentially closed models of the expanded theory are n-dimensional. In particular, a vector-space of dimension greater than n embeds in a space of dimension n so as to preserve independence in all n-element sets of vectors. This might be compared with the observation that existentially closed field-extensions have transcendence-degree one.

MATHEMATICS DEPARTMENT

MIDDLE EAST TECHNICAL UNIVERSITY

ANKARA 06531, TURKEY

E-mail: dpierce@metu.edu.tr

URL: http://www.math.metu.edu.tr/^{∼}dpierce/