Find all ring maps $\mathbb{Q}[x]/(x^{101}+2)\longrightarrow\mathbb{Q}[x]/(x^{501}-2)$

Question

I know that since $x^{101}+2$ and $x^{501}-2$ are both irreducible via Eisenstein, we have that $\mathbb{Q}[x]/(x^{101}+2)$ and $\mathbb{Q}[x]/(x^{501}-2)$ are actually fields. Since the kernel of any ring map is an ideal, any possible ring map will either be injective or trivial. So we are really looking for embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{Q}(\beta)$ where $\alpha,\beta$ are roots of $x^{101}+2$ and $x^{501}-2$ respectively. Any embedding will be a degree $101$ subextension of $\mathbb{Q}(\beta)$. By the Degree Formula, $$ 501=[\mathbb{Q}(\beta):\mathbb{Q}]=[\mathbb{Q}(\beta):\mathbb{Q}(\alpha)][\mathbb{Q}(\alpha):\mathbb{Q}]=[\mathbb{Q}(\beta):\mathbb{Q}(\alpha)]\cdot101, $$ which implies that $101$ divides $501$, a contradiction. Thus the only possible ring map is the trivial map. Is this solution correct?