Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded in the ring of integers $\mathcal{O}_K$ of a $p$-adic field $K$ for all but finitely many primes $p$.
(1) Is there a prime $p$ such that $R$ be embedded in $\mathcal{O}_K$ where $K/\mathbb{Q}_p$ is unramified? (2) If so, are there infinitely many such primes p?
I don't know how Cassels proved it, but it doesn't seem difficult from a geometric perspective. First, by "generic smoothness", replacing $R$ with a localization $R[1/f]$ we may assume it is smooth over $\operatorname{Spec}\mathbf{Z}$. Then, by some variant of Chebotarev's density theorem, for infinitely many primes $p$ there exists an $\mathbf{F}_p$-point, i.e. a homomorphism $\chi\colon R\to\mathbf{F}_p$. Now, $R$ embeds into its completion $\hat{R}_\chi$ with respect to the kernel of $\chi$, which is isomorphic to $\mathbf{Z}_p[[x_1, \ldots, x_n]]$ for some $n\geq 0$ (and we may take $x_1,\ldots,x_n\in R$). We now seek a continuous surjection $f\colon \hat{R}_\chi\to \mathbf{Z}_p$ whose restriction to $R$ is injective. Such a surjection is given by a tuple $a_1, \ldots, a_n$ of elements of $p\mathbf{Z}_p$. If a nonzero element of $R$ is in the kernel, it implies that the $a_i$ satisfy a polynomial equation with coefficients in $\mathbf{Q}$. It is therefore enough to choose the $a_i$ to be algebraically independent over $\mathbf{Q}$, which we can do since $\mathbf{Z}_p$ is uncountable.
