Suppose $K$ is a finite field extension of the p-adic numbers $\mathbb{Q}_p$. Let $T$ be the algebraic torus over $\mathbb{Q}_p$ obtained by the Weil Restriction of scalars from $K$ to $\mathbb{Q}_p$ of the multiplicative group $\mathbb{G}_m$ i.e. Res$_{K/\mathbb{Q}_p}(\mathbb{G}_m)$. Suppose $U$ is a non-empty open set of $K^*=T(\mathbb{Q}_p)$ in the $p$-adic topology on $K^*$. Then why is $U$ Zariski-dense in the torus $T$?
This came up in chapter III of Serre's Abelian $l$-adic Representations and Elliptic Curves, right after the definition of locally algebraic $p$-adic representations. Any help would be appreciated!
Lemma: Let $X$ be an irreducible variety over $\Bbb Q_p$. Then any open subset $U$ of $X(\Bbb Q_p)$ containing a smooth $\Bbb Q_p$-point is set-theoretically dense as a subset of $X$.
Proof: If $U$ is not set-theoretically dense in $X$, then it's contained in a proper closed subscheme $V$ of codimension one. By the implicit function theorem, any smooth point of $U$ has a basis of neighborhoods diffeomorphic to $\mathcal{O}^{\dim X}$. But this contradicts the fact that any smooth point of $V$ has a basis of neighborhoods diffeomorphic to $\mathcal{O}^{\dim X -1}$: two $p$-adic balls $\mathcal{O}^m$ and $\mathcal{O}^n$ are diffeomorphic iff $m=n$.
As $T$ is a smooth irreducible variety over $\Bbb Q_p$, this lemma applies to your situation and we're finished.