I haven't been able to find a 'multiple root version's of the Hensel's Lemma so below is my attempt at proving it using the generalized form of Hensel's Lemma:
Statement: Let $A$ be a complete discrete valuation ring and uniformizer $\pi$. Let $k = A/ \mathfrak{m}$ be it's residue field. Let $f(x) \in A[x]$ be such that its image in $k[x]$, $\bar{f}$ has a root $a_0$ of multiplicity $e$. Then there exists a unique root $a$ of $f$ with $a \equiv a_0 \pmod{\pi^e}$.
Proof: We have, $\bar{f}(x) = (x-a_0)^e h_0(x)$ where $h_0(a_0) \neq 0$ in $k$. By the general version of Hensel's Lemma, we can write $ f(x) = g(x) h(x)$ where $\bar{g} = (x-a_0)^e, \bar{h} = h_0$. We note that deg$(g)= e$.
Let $\alpha \in A$ such that $\bar{\alpha} = a_0 \pmod{\pi}$. Then $\alpha = a_0+a_1 \pi + a_2 \pi^2 + ... $ I'm not sure how to proceed from here. So I'd appreciate if anyone can help me on this.
Also, is the statement I've written above correct?
Thank you for any help.
