# A question on extension of valuations / absolute values

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!

• When you typed "extend this extension", did you mean "extend this absolute value"? Dec 3, 2020 at 3:33
• @J.W.Tanner oh you are right, I should write it more precisely. Dec 3, 2020 at 5:43

Your issue is that the fact you are recalling, i.e. that $$|\cdot|_K$$ has a unique extension to $$L$$, applies when $$K$$ a complete valued field (or more generally a "henselian" valued field, but I suspect the former is where you saw it come up). So the comment will not apply to $$K$$ a global field.