A single-segment Newton polygon implies henselian? I have a question about Newton polygons and henselian fields.
In p149 of Neukirch’s book(algebraic number theory:the beginning of Proposition 6.7), he says that
“We have just seen that the property of $K$ to be henselian follows from the condition that the Newton polygon of every irreducible polynomial $f(x)\in K[x]$ is a single segment.”
Question: In which place, this fact is shown?  Is it in the proof of 6.6? This fact is correct?
In Proposition 6.4, he showed that the uniqueness of extended valuations implies the condition of Newton polygons.
In Proposition 6.6, he showed that the equivalence between the uniqueness of extended valuations and henselianness.
However, I think that the proof of 6.6 uses not only the condition of Newton polygons but also the uniqueness(which I cannot remove).
On the other hand, there exists a monic irreducible polynomial such that its Newton polygon is a single segment but the splitting field of it has two extended valuations. (By an example $f(x)=x^2+1\in Q[x]$  and $p=5=(2+i)(2-i)$ given by Hagen at stackexchange/862893.)

Where do you see a problem?
Theorem 6.6 says that a field is henselian (that is Hensel's Lemma holds) if and only if the valuation extends uniquely to every algebraic extension.
The proof of the implication $\Rightarrow$ is already given in Theorem 6.2.
For the proof of the implication $\Leftarrow$ one assumes uniqueness, in which case the Newton polygon of an irreducible polynomial is a line. The latter is a consequence of Satz 6.4.
OK, I see now that the proof of Satz 6.7 really starts a bit in a confusing way. The point here is, that he shows that EVERY irreducible polynomial has a Newton polygon with only one segment, which by the remark three pages before is equivalent to the fact, that all roots possess the same value taking any extension of the valuation to the splitting field.
Now if for some algebraic extension of $K$ there are two different extensions of $v$, then one can find an irreducible polynomial having two roots with different values with respect to one of the two extensions of $v$.
That's it.
