I have a question about a comment occurred in the following former thread from MO:
https://mathoverflow.net/questions/309609/cohen-structure-theorem-with-explicit-equations
With the same setting as in the linked thread the questioner wanted to know how to construct the isomorphism
$$k[[t]][\alpha] \xrightarrow{\sim} k[[u]], \text{ where } \alpha^2 + a\alpha + b = 0$$
for general case.
According to @Mohan's comments one can reduce the problem to the case that $a=0$ and $b =t^mu$ with unit $u$.
What I don't understand is how Hensel's Lemma imply that $u$ has the shape $u = u(0)v^2$ for another unit $v$ with $v(0) =1$.
Indeed the equation become after the reduction step as above the shape $\tilde{\alpha}^2 +b=0$. But $\alpha \not \in k[[t]]$, so I don't understand why $v^2 \vert u$ and why $v \in k[[t]]$?
But I don't understand how exactly here Hensel was applied.
Remark: I'm working with following version of Hensel's lemma: https://en.wikipedia.org/wiki/Hensel%27s_lemma
So essentially it states that a coprime factorisation of a polynomial over residue field coeficients can be pulled back.