Let $K$ be a local field (e.g. a finite extension of $\mathbb{Q}_p$). Let $K^{ab}$ and $K^{ur}$ denote the maximal abelian and unramified extensions of $K$ inside an algebraic closure $\overline{K}$ of $K$ respectively. Is $K^{ab}$ a finite extension of $K^{ur}$?
Also, let $\theta_K$ be the reciprocity map of $K$. Also, let $\mathcal{O}_K$ denote the valuation ring of $K$. Is it true that $\theta_K(\mathcal{O}_K)\subseteq Gal(K^{ab}/K^{ur})$?