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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 $\theta_K(\mathcal{O}_K)\subseteq Gal(K^{ab}/K^{ur})$?

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up vote 1 down vote accepted

The answer to both questions is "no".

For the first question: $\mathbb{Q}_p(\sqrt[p^n]{1})/\mathbb{Q}_p$ is completely ramified, of degree $p^{n-1}(p-1)$. As a result, the ramification degree of an abelian extension of $\mathbb{Q}_p$ can be arbitrarily large, so $\mathbb{Q}_p^{ab}/\mathbb{Q}_p^{ur}$ is infinite.

For the second question (and say again $K=\mathbb{Q}_p$): $\theta(p)$ is the Frobenius automorphism, so it certainly acts non-trivially on $K^{ur}$.

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