  Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and
$\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$.
We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of
$E$ over the fixed subfield of $\bar{\bbq}$ under $\sigma$ has infinite rank.
Our approach uses the modularity of $E/\bbq$ and a collection of algebraic
points on $E$ -- the so-called {\em Heegner points} -- arising from the theory
of complex multiplication.
