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\begin{center}{\huge Fields and Algebraic Groups}\end{center}
\begin{center}{\large Katrin Tent (Wuerzburg)}\end{center}
\begin{center}{\large (joint with L. Kramer)}\end{center}

\bigskip

We show how the field of definition, $k$, of 
an isotropic absolutely almost simple $k$-group $G$
may be defined in the group $G(k)$ of $k$-rational points
in pure group theoretic language.
Our construction is inspired by the famous paper of Borel-Tits
\cite{BT3}.
However, since we are not dealing with homomophisms, we need
much less of the heavy machinery used by Borel-Tits.
The construction is as follows: inside the center of the unipotent 
radical of a minimal parabolic $k$-subgroup we pick a suitable
1-dimensional $k$-vector space. This defines the additive
group structure of $k$; the action of the center of a Levi $k$-subgroup
$L$ induces the multiplicative structure. Here we make use of
a result of Azad-Barry-Seitz \cite{ABS} concerning the action of
$L$.

This answers a question by
Borovik-Nesin \cite{BN} p. 367 (under the additional assumption 
that $G$ is $k$-isotropic). Easy examples also show that if $G$
is not absolutely simple, then $G(k)$ might be a simple group of 
finite Morley rank without $k$ being algebraically closed since $k$ 
might be the 'wrong' field of
definition. In most such cases one can replace $G$ by an
absolutely simple group $H$ with $H(l)\cong G(k)$ for some suitable 
field extension $l$ of $k$ - thus $l$ has to be algebraically closed.

\begin{thebibliography}{99}

\bibitem{ABS}
H. Azad, M. Barry, G.Seitz,
On the structure of parabolic subgroups,
Comm. Algebra 18, 551--562 (1990)

\bibitem{BT3}
A. Borel, J. Tits,
Homomorphismes ``abtraits'' de groupes alg\'ebriques simples,
Ann. Math. 97, 499--571 (1973)

\bibitem{BN}
A. Borovik, A. Nesin,
{\em Groups of finite Morley rank,}
Oxford Science Publ. 1994

\end{thebibliography}

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