In Neukirch's Class Field Theory, he promised that the local class field theory developed in the book is valid for fields of characteristic $0$ with a Henselian valuation, that is, an absolutely indecomposable discrete valuation with finite residue field. But I failed to find the definition of absolutely indecomposable discrete valuation. Is there any reference?
In Neukirch's another book Algebraic Number Theory he defines a Henselian valuation of $A_k$ to be
a homomorphism $v:A_k\to\hat{\mathbb{Z}}$ such that $v(A_k)=Z\supseteq\mathbb{Z}$, $Z/nZ\cong\mathbb{Z}/n\mathbb{Z}$ for all $n\in\mathbb{N}$ and $v(N_{K/k}A_v)=f_KZ$ for all finite extensions $K/k$.
I am not sure whether these two Henselian valuations mean the same thing, especially because in Class Field Theory the theory is developed by homological method, but not in the book Algebraic Number Theory (and these two definitions don't look alike at all).
Moreover, the most common definition of Henselian valuation I've seen is a valuation that extends uniquely to any algebraic extension of its quotient field. In my opinion, the homological method in the book is valid for this Henselian valuation that is discrete, characteristic $0$, and admits a finite residue field. So, am I right?