p-adic topology on finite extensions of $\mathbb{Q}_p$

Question

Let's take $\mathbb{Q}_p \subset K$ a finite extension endowed with the $p$-adic topology. For $z \in K$ is the application $x \to x+z $ continuous ? Can a set can be open and not be $K$ ? I actually want to show that given an open subgroup $I \subset O_K^{\times}$ the $\mathbb{Q}_p$ space its generate in $K$ his the entire $K$. So first it is easy to show that $<I>$ is open but I don't know if I can conclude from here ?

Answer 1

As your group $I$ is open, it contains a neighbourhood of $1$ and so a set of the form $1+\mathfrak{p}^n$ where $\mathfrak p$ is the maximal ideal of the valuation ring of $K$. Then the $\Bbb Z$-linear span of $I$ contains $\mathfrak{p}^n$ and so the $\Bbb Q$-linear span contains $\Bbb Q\mathfrak{p}^n$ which equals $K$.