Increase in rank of elliptic curves

Question

I expect answers to these questions are known, or at least partial answers are known:

1. Let $E$ be a rank 0 elliptic curve defined over $\mathbb{Q}$ and let $p$ be an odd prime. Is it possible that the rank of $E$ does not go up in any degree $p$ extension?
2. Are there rank 0 elliptic curves, for which the rank always goes up in a degree $p$ extension ($p\neq 2$)?

Answer 1

Expanding on Chris's answer, let $E/K$ be an elliptic curve defined over a number field. If you embed $E$ using the linear system $|n(O)|$ with $n\ge3$, you'll get $E$ as a smooth curve of degree $n$ in $\mathbb P^{n-1}_K$. Taking the intersection of $E$ with a generic hyperplane $H$ defined over $K$, for most choices of $H$ (in a Hilbert irreducibility sense), you should get $E\cap H=\{P_1,\ldots,P_n\}$ with $K(P_1)/K$ an extension of degree $n$, with $P_1$ non-torsion, and indeed, with $P_1$ independent from the points in $E(K)$. Filling in the details would prove:

Theorem Let $E/K$ be an elliptic curve defined over a number field. Then for every $n\ge1$ there exists an extension $L/K$ with $[L:K]=n$ and $\operatorname{rank}E(L)\ge\operatorname{rank}E(K)+1$.

Of course, this doesn't contradict Chris's second comment, since that refers to extensions $L/K$ that are Galois with cyclic Galois group.