Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree

Let $K$ be a field of characteristic $0$ and $E\mathrm{/}K$ an elliptic curve over $K$. For a finite extension $L\mathrm{/}K$ and a prime~$\mathrm{\ell }$, we provide Galois-theoretic sufficient conditions on $L\mathrm{/}K$ under which $E\left(L\right)\left[{\mathrm{\ell }}^{\mathrm{\infty }}\right]=E\left(K\right)\left[{\mathrm{\ell }}^{\mathrm{\infty }}\right]$. For a non-Galois extension $L\mathrm{/}K$ of prime degree, we relate the growth of the ${\mathrm{\ell }}^{\mathrm{\infty }}$-torsion subgroup of $E$ under the base change $L\mathrm{/}K$ to the image of the mod-$\mathrm{\ell }$ cyclotomic character. In particular, In particular, we refine Gonz{á}lez-Jim{é}nez’s result by ruling out certain torsion structures for quintic non-Galois extensions $L\mathrm{/}\mathbb{Q}$.