I'm reading "Fourier analysis on number fields" by Ramakrishnan and Valenza. Now I come to page 160: extension of absolute values. In the text the authors write: Let $K/L$ be a finite extension of global fields of degree $n$ (say number fields). Let $\mathscr{P}(K),\mathscr{P}(L)$ be the set of their places, respectively. An absolute value on $K$ gives rise to an absolute value on $L$ via restriction, so we obtain a map:
$r:=r_{K/L}:\mathscr{P}(K)\rightarrow \mathscr{P}(L), v\mapsto u.$
If $r(v)=u$, then $v$ is said to lie over $u$. And then there's a series propositions talking about the analysis of the fibers of this $r$.
What made me puzzled is: I've already known the basic fact on the extension of absolute values: i.e. the following theorem---If $K/L$ is an algebraic extension and $L$ is equipped with an absolute value $|\cdot|_K$, then there's a unique way to extend this extension to $K$, namely the formula $|\alpha|:=|N_{L/K}(\alpha)|_K,\forall \alpha \in L$. So I once thought that for an extension of number fields $L/K$ and a fixed place $u$ of $K$, there's a unique way to extend the absolute value to $(L,v)$ and obtain a unique extension of absolute values over local fields $L_v/K_u$. But according to the book I believe there's something wrong here: there are different places lying over $u$ in general.
Can anyone tell me what's wrong in my understanding of these two facts? Thanks in advance!
