  Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an
elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a
$K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each
$\sigma\in Gal(\bar{K}/K)$, the Mordell-Weil group $E(\bar{K}^{\sigma})$ of $E$
over the fixed subfield of $\bar{K}$ under $\sigma$ has infinite rank.

Number Theory