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Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, where $K_p^*$ is included in $\mathbb A_K^{*}$ by the map $x \mapsto (x,x,x,...)$ . $\psi_K:\mathbb A_K^{*} \to Gal(K^{ab}/K)$. Then there is Artin reciprocity law.

1) $\ker(\psi_K)=K^*$

2) $Im(\psi_K)$ is dense.

My question can it easily be deduced from local case?

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    $\begingroup$ The image can be characterised very explicitly: it consists of all those automorphisms that act as an integer power of Frobenius on the residue field $\bar{\mathbb{F}}_p$ of $K^{ab}$. Both statements are genuinely global. To give you an idea: class field theory of local function fields is very very similar to class field theory of $p$-adic fields. But the analogues of the two statements you are quoting are different for number fields: the kernel is bigger, whereas the image is actually all of $Gal(K^{ab}/K)$. $\endgroup$
    – Alex B.
    Commented May 18, 2014 at 10:34
  • $\begingroup$ @AlexB. Quite illuminating! Can you give a reference for an extensive treatment of all related topics in such combination and style? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 19, 2014 at 7:18

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