# Base-Change for $GL_1$

I try to understand the base-change in the theory of automorphic forms for the simplest case: $GL_1$.

Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of distinct Hecke characters of $\mathbb A_{K}^{\times}$ associated to this extension by class-field-theory. Now given a Hecke character $\chi$ of $\mathbb A_{K}^{\times}$, we define its base-change to $\mathbb A_{L}^{\times}$ by $\chi_L:=\chi \cdot \mathbb N_{L/K}$. Then one should check that: $$L(s,\chi_L)=\Pi _{\omega\in\Gamma }L(s,\chi\omega)$$

Conversely, suppose $L/K$ is cyclic with $Gal(L/K)=\langle\sigma\rangle$. Let $\chi^\prime$ be a Hecke character of $\mathbb A_L^\times$ with $\chi^\prime=\chi^\prime\cdot\sigma$, then one might expect that $$\chi^\prime=\chi_L$$ for some Hecke character of $\mathbb A_K^{\times}$.

My question is: How to prove these two results ? Or tell me some reference.

Thank you very much.

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I think it is considered not appropriate to simultaneously post the same question here and on Math Overflow. –  paul garrett Aug 10 '12 at 14:06
Sorry about this. –  user34327 Aug 10 '12 at 16:45
Certainly not without point out that you did so; that's an unnecessary duplication of efforts. –  joriki Aug 10 '12 at 16:52
For what it's worth, here's the link to the MO thread: mathoverflow.net/questions/104397/… –  M Turgeon Aug 10 '12 at 17:03
Dear all, please tell me wheather I should delete one of them. –  user34327 Aug 10 '12 at 18:20