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Reword title to ask the question asked below in a concise manner; explain HCF in body of text.
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edit approved Jan 11, 2016 at 14:58
Alex
edit approved Jan 11, 2016 at 14:58
Alex
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maximal unramified extension If $K=\langle$HCF of $\mathbb{Q}_{p}$ with $p$-power, $\mathbb{Q}^\mathrm{cycl}\rangle$, does $K$ also contain all roots of unityelements of $\mathcal{O}_{K}^{\times}$?

Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the Hilbert Class Field (i.e., maximal unramified extension) of $\mathbb{Q}_{2}^{\mathrm{unr}}$$\mathbb{Q}_{2}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.

maximal unramified extension of $\mathbb{Q}_{p}$ with $p$-power roots of unity

Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the maximal unramified extension $\mathbb{Q}_{2}^{\mathrm{unr}}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.

If $K=\langle$HCF of $\mathbb{Q}_{p}$, $\mathbb{Q}^\mathrm{cycl}\rangle$, does $K$ also contain all roots of elements of $\mathcal{O}_{K}^{\times}$?

Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the Hilbert Class Field (i.e., maximal unramified extension) of $\mathbb{Q}_{2}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.

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Jeff Yelton
Jeff Yelton
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maximal unramified extension of $\mathbb{Q}_{p}$ with $p$-power roots of unity

Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the maximal unramified extension $\mathbb{Q}_{2}^{\mathrm{unr}}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.