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Recently, I consider a problem: Let $E/F$ is a Galois extension of number field, denote the idele group of $E$ (resp $F$) by $I_E$ (resp $I_F$). There is a homomorphism induced by norm map $N_{E/F}$:

$$N: I_E/E^{\ast} \to I_F/F^{\ast}$$

If there is a finite abel extension $K/E$ ,from Global Class Field Theory,there is an isomorphism called Artin map:

$$A_K/E: I_E/E^{\ast} \text{Nm}(I_K) \to \text{Gal}(L/K)$$

If there is a subset $U$ of $I_E/E^{\ast}$ s.t $A_K/E(U) = 1$, where $1$ is the identity element of $\text{Gal}(K/E)$, So i really want know is there a special relationship between $U$ and the set $\text{Ker}(I_E/E{\ast} \to I_F/F^{\ast})$?

Thanks!

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Fixed some formatting. For future reference, it's a bad idea to use * for stars because these are interpreted as italics. –  Qiaochu Yuan Nov 27 '10 at 12:52

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