# Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$, are those that are lifted from $A$ by Hensel's Lemma.

Suppose that $b$ is a solution in $\mathbb{Z} / p^k \mathbb{Z}$ such that $b \not\equiv a \mod p$ with $a \in A$. It holds that $f(b) = 0 + n \cdot p^k$. Therefore it also holds that $f(b) = 0 + q \cdot p$. We can thus say $f(b) = 0 \mod p$. Because of this $\bar{b} \in A$. But this is impossible since there is no element in $A$ which is congruent to $b \mod p$. Hence the solutions lifted by Hensel's lemma are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$.

Is this a correct proof? If not could you give a counterexample? How would one then find the solutions for $\mathbb{Z} /p^k \mathbb{Z}$ in function of $k$?

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