Henselization and immediate extension

Question

I am reading about the henselization and immediate extension of valuation. I am getting confusion about some basic terminology. I have few question. \ $1)$ Is every hensilization extension of valuation is immediate or not?\ 2) Is every extension of hensel valued field is immediate extension or not?\ 3) Is every rank $1$ valuation satisfies hensel lemma? Thanks

Answer 1

I am not used to work with arbitrary valuation fields, but a quick googling leads to a paper here, its Theorem 1.1 answers positively your question (1).

(2) and (3) are false already for discrete valuations. For (2): any complete DVD is henselian and has ramified extensions (adjoin a square root of an uniformizing element); For (3) the $p$-adic valuation on $\mathbb Q$ is rank 1, but is absolutely not henselian: $T^2-(1+p)\in \mathbb Z[T]$ is split mod $p>2$, but it has no root in $\mathbb Q$.