%Abstract of contributed talk by David Pierce, LC 2006, Nijmegen

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\title{Vector-spaces over unspecified fields}

\author{David Pierce}
\revauthor{Pierce, David}

\address{Mathematics Department\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/\urltilde dpierce/}

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In the \emph{Elements,} Euclid of Alexandria gives a geometric
formulation of certain field-theoretic identities; for example, the
identity
\begin{equation*}
  x^2-y^2=(x+y)(x-y)
\end{equation*}
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 \emph{Geometry,} Ren\'e{} Descartes shows how lines can be
multiplied to produce \emph{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 \emph{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.

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