Hensel's lemma & $p$-adic polynomial roots

Question

I want to determine the number of roots of $f(X) = X^3-5X+20$ in $\mathbb{Z}_p$ using Hensel's lemma (lemma is on the bottom). Unfortunately I am not very well trained to solve this. Take for example $p=2$. Then reducing the polynomial modulo 2 we get \begin{align*} f(X) &= X^3 - 5X+20 \equiv 0 \mod 2 \\ f'(X) &= 3X^2-5\equiv X + 1 \mod 2 \end{align*} That would mean that only $0\in\mathbb{Z}/2\mathbb{Z}$ gives a solution such that $f(0) \equiv 0$ and $f'(0)\not \equiv 0$ modulo 2, which then by Hensel's lemma gives a unique root of $f$ in $\mathbb{Z}_2$, which is $0$ modulo $2$.

The problem is that I don't know the relation between finding a root via Hensel's lemma and there really being one. I have found one root via Hensel's lemma, but what guarantees me there aren't more?

Any help on how to proceed is most welcome. Thank you in advance.

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The lemma I am using:

Let $K$ be a complete field with discrete valuation $v$, valuation ring $O$ and residue field $k$. Let $f(X)\in O(X)$. Suppose that the reduction $\bar{f}(X)\in k[X]$ has a simple root $\bar{a}$, ie $\bar{f}(\bar{a}) = 0$ and $\bar{f}'(\bar{a})\neq0$. Then there exists a unique $x\in O$ such that $f(x) = 0$ and $\bar{x}=\bar{a}$.

Answer 1

This polynomial has a double root at $1$ in $\mathbb{F}_2$, so the out of the box Hensel's lemma won't apply. However we can still apply a modified version by going one power of two up. In particular the double root at 1 splits into two simple roots mod 4 (at 1 and 3). Now a modified Hensel's lemma (with essentially the same proof as the usual version) gives us that these each have a unique 2-adic lift.

One thing that is different about this modified Hensel's lemma from what you are used to is that if we want to solve $f(x) \equiv 0 \bmod 2^n$, the lifts we get will only be unique mod $2^{n-1}$. In particular we get that $f(x) \equiv 0 \bmod 2^n$ has 5 solutions for all $n > 2$.

Answer 2

Just for the number of roots in the $2$-adic integers $\Bbb Z_2$, I recommend forming $g(X)=f(X+5)=X^3+15X^2+70X+120$. Then I use something rather more powerful than plain old Hensel, namely the Newton polygon, in which you plot $(n,v_2(a_n))$ for every monomial $a_nX^n$ in your polynomial. The points you plot then are $(0,3)$, $(1,1)$, $(2,0)$, and $(3,0)$. You then take the “upper convex hull” of these points, which has segments of slope $-2$, $-1$, and $0$, each of width $1$. This says that in $\Bbb Z_2$, there are roots of $2$-adic valuation $2$, $1$, and $0$, three in all. You may be able to take $g$ and find the roots directly with your form of Hensel.