Hensel's lemma is the \(p\)-adic analogue of Newton's method and the implicit-function theorem. A simple solution modulo \(p\) lifts uniquely to solutions modulo every \(p^n\) and hence to a root in \(\mathbb Z_p\).
Simple-root lifting
Theorem (Hensel). Let \(f\in\mathbb Z_p[x]\). If \(a_1\in\mathbb Z\) satisfies \[ f(a_1)\equiv0\pmod p,\qquad f'(a_1)\not\equiv0\pmod p, \] then there is a unique \(a\in\mathbb Z_p\) such that \[ f(a)=0,\qquad a\equiv a_1\pmod p. \]
Proof. Suppose \(f(a_n)\equiv0\pmod{p^n}\). Seek \(a_{n+1}=a_n+t_np^n\). Taylor expansion gives \[ f(a_{n+1}) \equiv f(a_n)+t_np^nf'(a_n)\pmod{p^{n+1}}. \] Because \(f'(a_n)\) is a unit modulo \(p\), there is a unique \(t_n\bmod p\) making the right side zero. Thus \(a_{n+1}\equiv a_n\pmod{p^n}\), so \((a_n)\) is Cauchy and converges to a root. The same linear congruence proves uniqueness at each level and hence in \(\mathbb Z_p\).
Example. Solve \(x^2=2\) in \(\mathbb Z_7\). Since \(3^2\equiv2\pmod7\) and \(2\cdot3\not\equiv0\), there is a unique root congruent to \(3\). To lift modulo \(49\), write \(x=3+7t\): \[ x^2-2\equiv7+42t=7(1+6t)\pmod{49}, \] so \(t\equiv1\pmod7\) and \(x\equiv10\pmod{49}\).
Newton form
Theorem. If \[ |f(a_0)|_p<|f'(a_0)|_p^2, \] then Newton iteration \[ a_{n+1}=a_n-\frac{f(a_n)}{f'(a_n)} \] is defined, converges quadratically in the valuation, and its limit is the unique root in the corresponding ball.
Proof outline. Taylor expansion and the ultrametric inequality show that \(f'(a_n)\) keeps the same absolute value and \[ \left|\frac{f(a_{n+1})}{f'(a_{n+1})^2}\right|_p \le \left|\frac{f(a_n)}{f'(a_n)^2}\right|_p^2. \] Thus increments tend to zero geometrically in valuation, making the sequence Cauchy. The same estimate applied to two roots in the ball gives uniqueness.
Factorization form
Hensel factorization lemma. If a monic polynomial satisfies \[ \overline f=\overline g\,\overline h\quad\text{in }\mathbb F_p[x], \qquad\gcd(\overline g,\overline h)=1, \] then there are unique monic \(g,h\in\mathbb Z_p[x]\), reducing to the given factors, such that \(f=gh\).
This form lifts a coprime factorization rather than one root. It is central to decomposing primes in local extensions and to algorithms for factoring polynomials over \(\mathbb Q_p\).
Local solvability and its limits
Hensel's lemma turns a nonsingular solution over the residue field into a \(p\)-adic one. Singular roots need separate analysis: \(f'(a)\equiv0\pmod p\) may allow no lift, several lifts, or a whole \(p\)-adic family. Local solvability for every \(p\) and over \(\mathbb R\) is necessary for a rational solution, but not sufficient for arbitrary Diophantine equations.