Let $E$ be an elliptic curve over a finitely generated infinite field $K$. Let $K^{\rm s}$ denote a separable closure of $K$, $\sigma$ an element of the Galois group $G_{K}\,{=}\,\hbox{Gal}(K^{\rm s}/K)$, and $K^{\rm s}(\sigma)$ the invariant subfield of $K^{\rm s}$. If the characteristic of $K$ is not 2 and $\sigma$ belongs to a suitable open subgroup of $G_K$, then $E(K^{\rm s}(\sigma))$ has infinite rank.Partially supported by the Sloan Foundation and by NSF Grant DMS 97-27553.