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.
