Let $E/\Bbb{Q}$ be an elliptic curve.
Rank of elliptic curve over quadratic extension $L=K(\sqrt{D})/K$ is calculated by a formula $rank(E/L)=rank(E/K)+rank(E_D/K)$(this is easy to prove). In particular, rank of elliptic curve does not decrease under quadratic field extension.
But when $L/K$ is a extension of degree $n\ge 3$, does $rank(E/L)\ge rank(E/K)$ always hold?
Yes. If the curve has rank $n$, there will be $n$ points on the curve, so that no nontrivial relation (in the sense of group presentations) holds among these points. Vice versa, the existence of $n$ such points implies that the rank is at least $n$.
But relations (since there is no scalar multiplication) do not depend on the field extension.