First question on MathOverflow, I hope it is appropriate for this site. There are two related questions.

Let $K$ be a number field, let $G_K = Gal(\overline{K}/K)$, and let 
$$\chi_1,\chi_2:G_K\rightarrow\overline{\mathbb{Q}}^\times$$
be Hecke characters such that $\ker(\chi_1) = \ker(\chi_2)$. 

My first question is, are the two fields generated by the values of $\chi_1$ and $\chi_2$ the same? That is, does 
$$\mathbb{Q}(\chi_1(\sigma), \sigma\in G_K) = \mathbb{Q}(\chi_2(\sigma),\sigma\in G_K)$$
As far as I can tell, this is true if $\chi_1,\chi_2$ have finite image, but I'm not sure if it's true if they have infinite image.

My second question is, does there exists $\sigma\in G_\mathbb{Q} = Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ such that for all $\tau\in G_K$, $\chi_1(\tau) = \sigma(\chi_2(\tau))$?

If $E = \mathbb{Q}(\chi_1(\sigma), \sigma\in G_K) = \mathbb{Q}(\chi_2(\sigma),\sigma\in G_K)$, then one could define $\sigma$ by the rule $\sigma(\chi_2(\tau)) = \chi_1(\tau)$, show $\sigma\in Gal(E/\mathbb{Q})$, and take an extension of $\sigma$ to an element of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Then the answer to the second question would be yes if the answer to the first question is yes.

I don't know how to show that $\sigma$ defined as above is a well defined element of $Gal(E/\mathbb{Q})$.