I am on Jurgen Neukirch' s Algebraic Number theory.
On page 131, theorem 4.8 says “Let $K$ be complete with respect to the valuation | |. Then | | may be extended in a unique way to a valuation of any given algebraic extension $L/K$.”
But on page 161, theorem 8.1 says “Let $L/K$ be an algebraic field extension $\mathcal{v}$ a valuation of K. Then one has: Every extension $\mathcal{w}$ of the valuation $\mathcal{v}$ arises as the composite $\mathcal{w}=\overline{\mathcal{v}} \tau$”, for some $K$-embedding $\tau :L \to \overline{K}_{v}$.
I am confused when the extension is unique, when is not unique? The second the theorem should also include the case in the first one. Is it because of the first theorem require completeness for $K$?
I have no idea what is the difference between these two theorems.
Here is the pdf version of that book:
http://www.cimat.mx/~luis/seminarios/Teoria-de-Numeros/Neukirch_Algebraic_number_theory.pdf
Thanks in advance!
Yes, as you said, the difference is that $K$ is not necessarily complete in the latter.
I strongly recommend you read the beginning couple of paragraphs to chapter 8 (and to all chapters!): these will give you an idea of the bigger picture. It says there explicitly that, having dealt with henselian fields, you're now going to study general fields; and if you look up the word "henselian" in the index, it sends you to chapter 6, whose introduction tells you that henselian fields are a generalisation of complete fields.