21.02.04 · number-theory / quadratic-forms-local-fields

Hensel's lemma

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Anchor (Master): Hensel 1897 *Jahresber. Deutsch. Math.-Verein.* 6, 83-88 (originator: factorisation lifting from $\mathbb{F}_p$ to $\mathbb{Z}_p$); Hensel 1908 *Theorie der algebraischen Zahlen* (Teubner, Leipzig — first systematic treatment); Krasner 1944 *J. Math. Pures Appl.* 23 (Krasner's lemma and the structure of extensions of $\mathbb{Q}_p$); Serre 1973 Ch. II §2; Cassels 1986 Ch. 4; Neukirch 1999 Ch. II §§4-6; Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 1 §3; Bosch-Güntzer-Remmert *Non-Archimedean Analysis* (Springer Grundlehren 261, 1984) §3.4 (Henselian valuation rings); Eisenbud *Commutative Algebra with a View toward Algebraic Geometry* (Springer GTM 150, 1995) §7.4 (Hensel's lemma in commutative-algebra framing)

Intuition Beginner

Hensel's lemma is the $p$ -adic version of Newton's method, and it answers a very concrete question: if you have an approximate solution to a polynomial equation modulo a prime $p$ , when can you sharpen it to an exact solution in the $p$ -adic integers? The lemma says: if your approximate root is good enough that the polynomial value at it is divisible by a higher power of $p$ than the derivative value is, you can always sharpen the approximation, step by step, and the steps converge.

Think of decimal expansions. If you know that $x^{2} = 2$ has a solution near $1.4$ in the real numbers, Newton's method refines $1.4$ to $1.41$ to $1.414$ to $1.4142$ , each step doubling the number of correct digits. Hensel's lemma is the same idea but for digits in base $p$ . If $x^{2} = - 1$ has a solution modulo $5$ (and it does: $2^{2} = 4 \equiv - 1$ ), you can lift $2$ to a solution modulo $25$ , then modulo $125$ , and so on, and the limit lives in the $5$ -adic integers $Z_{5}$ .

The reason this matters: it turns local solvability (one congruence modulo $p$ ) into honest existence (an actual solution in $Z_{p}$ ). Whole branches of number theory rest on this leverage, including the local-global principle, the theory of $p$ -adic representations, and the structure of the multiplicative group $Z_{p}^{\times}$ .

Visual Beginner

A schematic showing the Newton iteration in the $p$ -adic setting. Plot a vertical axis representing $p$ -adic distance: closer to zero means more digits of agreement. Mark $a_{0}$ on the right, then $a_{1}$ closer to the unknown root, then $a_{2}$ closer still, with each step halving the gap on a logarithmic scale (in the $p$ -adic sense, that means doubling the power of $p$ that divides the gap).

The picture captures the geometric content: each Newton step at least doubles the precision in the $p$ -adic absolute value. The Cauchy property of the sequence is what allows the limit to live inside the complete ring $Z_{p}$ .

Worked example Beginner

Compute a square root of $- 1$ in the $5$ -adic integers, step by step. The aim is to find $a \in Z_{5}$ with $a^{2} = - 1$ , starting from the observation that $2^{2} = 4 \equiv - 1 (mod 5)$ .

Step 1. Set $f (x) = x^{2} + 1$ . At the seed $a_{0} = 2$ the value is $f (2) = 5$ and the derivative value is $f^{'} (2) = 4$ . The $5$ -adic absolute value of $f (2)$ is $∣5 ∣_{5} = 1/5$ , and the $5$ -adic absolute value of $f^{'} (2) = 4$ is $∣4 ∣_{5} = 1$ . The Hensel hypothesis $∣ f (a_{0}) ∣_{5} < ∣ f^{'} (a_{0}) ∣_{5}^{2}$ reads $1/5 < 1$ , which holds.

Step 2. Apply the Newton update $a_{1} = a_{0} - f (a_{0}) / f^{'} (a_{0}) = 2 - 5/4$ . Working modulo $25$ : the multiplicative inverse of $4$ modulo $25$ is $19$ since $4 \cdot 19 = 76 = 3 \cdot 25 + 1$ . So $a_{1} \equiv 2 - 5 \cdot 19 = 2 - 95 \equiv 2 - 95 + 4 \cdot 25 = 7 (mod 25)$ . Check: $7^{2} = 49 = 2 \cdot 25 - 1$ , so $7^{2} \equiv - 1 (mod 25)$ . The lifted approximation works.

Step 3. Repeat to obtain a solution modulo $125$ . The Newton update gives $a_{2} = a_{1} - f (a_{1}) / f^{'} (a_{1})$ . Compute $f (7) = 50$ and $f^{'} (7) = 14$ . The inverse of $14$ modulo $125$ is $9$ , since $14 \cdot 9 = 126 \equiv 1 (mod 125)$ . So $a_{2} \equiv 7 - 50 \cdot 9 = 7 - 450 \equiv 7 - 75 = - 68 \equiv 57 (mod 125)$ . Check: $5 7^{2} = 3249 = 25 \cdot 125 + 124$ , so $5 7^{2} \equiv - 1 (mod 125)$ .

Step 4. Continue. Modulo $625$ the lifted approximation is $a_{3} \equiv 182 (mod 625)$ , and modulo $5^{5} = 3125$ it is $a_{4} \equiv 2057 (mod 3125)$ . The sequence $a_{0}, a_{1}, a_{2}, a_{3}, \dots = 2, 7, 57, 182, 2057, \dots$ has the property that $a_{n}^{2} \equiv - 1 (mod 5^{n + 1})$ , and the $5$ -adic limit is one square root of $- 1$ in $Z_{5}$ . The other square root is the additive inverse, with sequence $3, 18, 68, 443, 1068, \dots$ .

What this tells us: Hensel's lemma is a constructive recipe. Each Newton step computes the next $p$ -adic digit, and the running approximation matches the true root modulo a higher power of $p$ at every stage. The $p$ -adic absolute value packages this digit-by-digit agreement as honest convergence in a complete metric space, and the limit is a genuine root.

Check your understanding Beginner

Formal definition Intermediate+

Let $p$ be a prime and let $Z_{p}$ denote the ring of $p$ -adic integers, the completion of $Z$ with respect to the $p$ -adic absolute value $∣ x ∣_{p} = p^{- v_{p} (x)}$ where $v_{p} (x)$ is the largest power of $p$ dividing $x$ , with $v_{p} (0) = + \infty$ . The absolute value is non-archimedean: $∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ , with equality when $∣ x ∣_{p} \neq = ∣ y ∣_{p}$ . The unit group is $Z_{p}^{\times} = {x \in Z_{p} : ∣ x ∣_{p} = 1}$ , the elements not divisible by $p$ . The reduction map $Z_{p} ↠ F_{p}$ has kernel $p Z_{p}$ , and for $f \in Z_{p} [X]$ we write $\overset{ˉ}{f} \in F_{p} [X]$ for the reduction modulo $p$ .

Hensel's lemma (Newton-iteration form). Let $f \in Z_{p} [X]$ and let $a_{0} \in Z_{p}$ satisfy $$ |f(a_0)|_p < |f'(a_0)|_p^2. $$ Then there exists a unique $a \in Z_{p}$ with $f (a) = 0$ and $∣ a - a_{0} ∣_{p} \leq ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}$ .

The strict inequality $∣ f (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}^{2}$ is what powers the quadratic convergence of the Newton iteration $a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})$ .

Hensel's lemma (factorisation-lifting form). Let $f \in Z_{p} [X]$ and suppose the reduction $\overset{ˉ}{f} \in F_{p} [X]$ factors as $\overset{ˉ}{f} = \overset{g}{ˉ}_{0} \overset{ˉ}{h}_{0}$ with $\overset{g}{ˉ}_{0}, \overset{ˉ}{h}_{0} \in F_{p} [X]$ coprime. Then there exist $g, h \in Z_{p} [X]$ with $f = g h$ , $\overset{g}{ˉ} = \overset{g}{ˉ}_{0}$ , $\overset{ˉ}{h} = \overset{ˉ}{h}_{0}$ , and $de g g = de g \overset{g}{ˉ}_{0}$ . The lift $(g, h)$ is unique once one normalises $g$ to be monic of the same degree as $\overset{g}{ˉ}_{0}$ .

Henselian local ring. A local ring $R$ with maximal ideal $m$ and residue field $k = R / m$ is Henselian when, for every monic $f \in R [X]$ and every coprime factorisation $\overset{ˉ}{f} = \overset{g}{ˉ}_{0} \overset{ˉ}{h}_{0}$ in $k [X]$ with $\overset{g}{ˉ}_{0}$ monic, there exist monic $g, h \in R [X]$ lifting the factorisation. Equivalently, every simple root of $\overset{ˉ}{f}$ in $k$ lifts to a root of $f$ in $R$ .

Counterexamples to common slips

The hypothesis $∣ f (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}^{2}$ is strict, not weak. Take $f (X) = X^{2} - p$ and $a_{0} = 0$ : then $f (a_{0}) = - p$ and $f^{'} (a_{0}) = 0$ , so the right-hand side is $0$ and the hypothesis fails. The square root of $p$ does not lie in $Z_{p}$ ; it generates a ramified quadratic extension. The vanishing of $f^{'} (a_{0})$ in $F_{p}$ is the failure point.
Coprimality of $\overset{g}{ˉ}_{0}$ and $\overset{ˉ}{h}_{0}$ in $F_{p} [X]$ is required for the factorisation lift. Take $f (X) = X^{2}$ and the factorisation $\overset{ˉ}{f} = X \cdot X$ : here $\overset{g}{ˉ}_{0} = \overset{ˉ}{h}_{0} = X$ share the factor $X$ . The lift is not unique, since $f = X^{2} = X \cdot X = (X + 0) (X + 0)$ is the only factorisation up to associates. Coprimality is what isolates the lift.
The polynomial $f$ must have coefficients in $Z_{p}$ , not $Q_{p}$ . For $f (X) = X^{2} - 1/ p$ the leading coefficient is fine but the constant term lives in $Q_{p} ∖ Z_{p}$ ; the Newton iteration leaves $Z_{p}$ at the first step. The natural setting is $f \in O [X]$ where $O$ is a complete discrete valuation ring; for ramified extensions, the discrete valuation on $O$ replaces the $p$ -adic valuation on $Z_{p}$ .

Key theorem with proof Intermediate+

Theorem (Hensel's lemma, Newton-iteration form; Serre Ch. II §2 Theorem 1). Let $f \in Z_{p} [X]$ and let $a_{0} \in Z_{p}$ satisfy $∣ f (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}^{2}$ . Then there exists a unique $a \in Z_{p}$ with $f (a) = 0$ and $∣ a - a_{0} ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}$ .

Proof. Set $c = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}^{2} < 1$ and define the Newton iteration $a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})$ , working entirely in $Q_{p}$ at first and then verifying that each $a_{n}$ lies in $Z_{p}$ . The argument has four steps: each Newton step stays in $Z_{p}$ , the sequence $(a_{n})$ is Cauchy in $Z_{p}$ , the limit is a root, and the root is unique.

Step 1: $a_{n} \in Z_{p}$ and the derivative stays a non-zero-mod- $p$ -square. The Taylor expansion of $f$ around $a_{n}$ reads $$ f(a_n + \varepsilon) = f(a_n) + \varepsilon f'(a_n) + \varepsilon^2 R(\varepsilon) $$ for some $R (ε) \in Z_{p} [ε]$ when $a_{n} \in Z_{p}$ . Take $ε = - f (a_{n}) / f^{'} (a_{n})$ , so that $a_{n + 1} = a_{n} + ε$ . The Hensel hypothesis at the $n$ -th step, $∣ f (a_{n}) ∣_{p} < ∣ f^{'} (a_{n}) ∣_{p}^{2}$ , gives $∣ ε ∣_{p} < ∣ f^{'} (a_{n}) ∣_{p} \leq 1$ , so $ε \in Z_{p}$ and $a_{n + 1} \in Z_{p}$ . The Taylor expansion gives $$ f(a_{n+1}) = f(a_n) + \varepsilon f'(a_n) + \varepsilon^2 R(\varepsilon) = \varepsilon^2 R(\varepsilon), $$ since the linear term cancels by definition of $ε$ . The size of the new value is bounded by $∣ f (a_{n + 1}) ∣_{p} \leq ∣ ε ∣_{p}^{2} = ∣ f (a_{n}) ∣_{p}^{2} /∣ f^{'} (a_{n}) ∣_{p}^{2}$ . The Taylor expansion of $f^{'}$ around $a_{n}$ , namely $f^{'} (a_{n + 1}) = f^{'} (a_{n}) + ε f^{''} (a_{n}) /2 + \dots$ , has its leading correction of size $∣ ε ∣_{p} \leq c ∣ f^{'} (a_{n}) ∣_{p} < ∣ f^{'} (a_{n}) ∣_{p}$ . By the strong ultrametric equality, $∣ f^{'} (a_{n + 1}) ∣_{p} = ∣ f^{'} (a_{n}) ∣_{p}$ . So the derivative magnitude stabilises.

Step 2: the sequence is Cauchy in $Z_{p}$ . Let $c = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}^{2} < 1$ and set $M = ∣ f^{'} (a_{0}) ∣_{p}$ , which is fixed across iterations by Step 1. Define $c_{n} = ∣ f (a_{n}) ∣_{p} / M^{2}$ . The bound from Step 1 reads $c_{n + 1} \leq c_{n}^{2}$ , hence $c_{n} \leq c^{2^{n}}$ . So $$ |a_{n+1} - a_n|_p = |\varepsilon|_p = |f(a_n)|_p / |f'(a_n)|_p = c_n M \leq c^{2^n} M. $$ The geometric bound $c^{2^{n}}$ goes to zero, and the sequence $(a_{n})$ is Cauchy in the ultrametric space $Z_{p}$ . Since $Z_{p}$ is complete, the sequence converges to a limit $a \in Z_{p}$ .

Step 3: the limit is a root. Continuity of $f$ on the complete metric space $Z_{p}$ gives $f (a) = lim_{n} f (a_{n})$ . The bound $∣ f (a_{n}) ∣_{p} \leq c^{2^{n}} M^{2} \to 0$ forces $f (a) = 0$ . The distance bound is $∣ a - a_{0} ∣_{p} = lim_{n} ∣ a_{n} - a_{0} ∣_{p} \leq max_{k \leq n} ∣ a_{k + 1} - a_{k} ∣_{p} \leq c M = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}$ , strictly less than $∣ f^{'} (a_{0}) ∣_{p}$ by the Hensel hypothesis.

Step 4: uniqueness. Suppose $a, a^{'} \in Z_{p}$ are both roots of $f$ with $∣ a - a_{0} ∣_{p}, ∣ a^{'} - a_{0} ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}$ . The ultrametric inequality gives $∣ a - a^{'} ∣_{p} \leq max (∣ a - a_{0} ∣_{p}, ∣ a^{'} - a_{0} ∣_{p}) < ∣ f^{'} (a_{0}) ∣_{p}$ . Expand $f$ around $a$ : $0 = f (a^{'}) = f (a) + (a^{'} - a) f^{'} (a) + (a^{'} - a)^{2} S (a^{'} - a)$ for some $S \in Z_{p} [ε]$ . The Taylor expansion of $f^{'}$ at $a_{0}$ and the equality $∣ f^{'} (a) ∣_{p} = ∣ f^{'} (a_{0}) ∣_{p}$ (from Step 1) give the equation $(a^{'} - a) (f^{'} (a) + (a^{'} - a) S (a^{'} - a)) = 0$ . The second factor has $p$ -adic absolute value $∣ f^{'} (a_{0}) ∣_{p}$ since the correction term has smaller absolute value, so the second factor is non-zero in $Q_{p}$ , forcing $a^{'} = a$ . $□$

Bridge. Hensel's lemma builds toward every existence theorem for $p$ -adic solutions of polynomial equations, and it appears again in 21.07.01 ( $Z_{p}$ -extensions and Iwasawa theory) as the constructive input that produces roots of unity inside cyclotomic towers. The foundational reason it works is exactly the ultrametric triangle inequality: in a non-archimedean setting, the quadratic-error term of Taylor expansion has $p$ -adic absolute value $∣ ε ∣_{p}^{2}$ , which is strictly smaller than $∣ ε ∣_{p}$ for $∣ ε ∣_{p} < 1$ , and the Newton iteration converges by a geometric-of-geometric rate. The central insight is that completeness of $Z_{p}$ plus the Hensel hypothesis is exactly what guarantees the Newton sequence is Cauchy and the limit a root. This is dual to the archimedean Newton's method, where convergence requires an extra hypothesis on the second derivative; the ultrametric setting makes the quadratic convergence automatic. The bridge is the recognition that Hensel's lemma generalises to any complete discrete valuation ring, identifies completeness with the property "every simple residual root lifts", and the abstract Henselian-ring formulation packages this exactly. Putting these together, Hensel's lemma is the structural property of $Z_{p}$ that distinguishes it from $Z$ : a residual approximation always lifts to a $p$ -adic solution, the foundational reason the local-global philosophy of number theory has any traction.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the group of units $Z_{p}^{\times}$ contains a cyclic subgroup of order $p - 1$ for every odd prime $p$ , by lifting the cyclic generator of $F_{p}^{\times}$ via Hensel's lemma.

Hint

Apply Hensel's lemma to the polynomial $f (X) = X^{p - 1} - 1$ . Each non-zero element of $F_{p}$ is a root.

Answer

Let $f (X) = X^{p - 1} - 1 \in Z_{p} [X]$ . The reduction $\overset{ˉ}{f} (X) = X^{p - 1} - 1 \in F_{p} [X]$ has every non-zero element of $F_{p}$ as a simple root by Fermat's little theorem. At any seed $a_{0} \in {1, 2, \dots, p - 1}$ , the derivative is $f^{'} (a_{0}) = (p - 1) a_{0}^{p - 2}$ , which reduces to $- a_{0}^{p - 2} \neq = 0$ in $F_{p}$ , so $∣ f^{'} (a_{0}) ∣_{p} = 1$ . The value is $f (a_{0}) = a_{0}^{p - 1} - 1 \equiv 0 (mod p)$ but $\neq = 0$ in $Z$ , so $∣ f (a_{0}) ∣_{p} \leq 1/ p < 1 = ∣ f^{'} (a_{0}) ∣_{p}^{2}$ . Hensel's lemma lifts each seed to a root of $f$ in $Z_{p}$ , giving $p - 1$ distinct $(p - 1)$ -th roots of unity. These form the Teichmüller subgroup $μ_{p - 1} (Z_{p}) \subset Z_{p}^{\times}$ , a cyclic group of order $p - 1$ providing a canonical splitting of the reduction map $Z_{p}^{\times} ↠ F_{p}^{\times}$ . Rubric: full credit for the application of Hensel plus identification with the Teichmüller subgroup.

Exercise 4 (medium, short-answer).

Determine which integers $n$ are squares in $Z_{5}^{\times}$ .

Hint

A unit $u \in Z_{5}^{\times}$ is a square if and only if its image in $F_{5}^{\times}$ is a square. Use Hensel's lemma on $f (X) = X^{2} - u$ .

Answer

A $5$ -adic unit $u \in Z_{5}^{\times}$ is a square in $Z_{5}$ if and only if its reduction $\overset{u}{ˉ} \in F_{5}^{\times}$ is a square. The squares in $F_{5}^{\times} = {1, 2, 3, 4}$ are ${1, 4}$ (since $1^{2} = 1$ , $2^{2} = 4$ , $3^{2} = 4$ , $4^{2} = 1$ ). So an integer $n$ coprime to $5$ is a square in $Z_{5}$ iff $n \equiv 1$ or $4 (mod 5)$ . The Hensel lift works because at any seed $a_{0}$ with $a_{0}^{2} \equiv u (mod 5)$ the derivative $f^{'} (a_{0}) = 2 a_{0}$ is a unit in $Z_{5}$ (since $5$ is odd and $a_{0}$ is a unit), so the hypothesis $∣ f (a_{0}) ∣_{5} < ∣ f^{'} (a_{0}) ∣_{5}^{2}$ reads $1/5 < 1$ . For general $n = 5^{k} u$ with $u \in Z_{5}^{\times}$ , $n$ is a square in $Q_{5}$ iff $k$ is even and $\overset{u}{ˉ} \in {1, 4}$ . Rubric: full credit for the residue-class characterisation plus the Hensel-derived lift.

Exercise 5 (medium, short-answer).

State and prove the factorisation-lifting form of Hensel's lemma in the special case where $\overset{ˉ}{f} = (X - \overset{α}{ˉ}) \overset{ˉ}{h}$ with $\overset{α}{ˉ} \in F_{p}$ a simple root.

Hint

A simple root $\overset{α}{ˉ}$ of $\overset{ˉ}{f}$ means $\overset{ˉ}{f}^{'} (\overset{α}{ˉ}) \neq = 0$ . Reduce to the Newton-iteration form applied to a lift $a_{0} \in Z_{p}$ of $\overset{α}{ˉ}$ .

Answer

Let $\overset{α}{ˉ} \in F_{p}$ be a simple root of $\overset{ˉ}{f}$ , meaning $\overset{ˉ}{f} (\overset{α}{ˉ}) = 0$ and $\overset{ˉ}{f}^{'} (\overset{α}{ˉ}) \neq = 0$ . Choose any lift $a_{0} \in Z_{p}$ of $\overset{α}{ˉ}$ ; then $f (a_{0}) \equiv 0 (mod p)$ so $∣ f (a_{0}) ∣_{p} \leq 1/ p$ , and $f^{'} (a_{0})$ reduces to $\overset{ˉ}{f}^{'} (\overset{α}{ˉ}) \neq = 0$ in $F_{p}$ so $∣ f^{'} (a_{0}) ∣_{p} = 1$ . The Hensel hypothesis $∣ f (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}^{2}$ reads $1/ p < 1$ , which holds. The Newton-iteration form lifts $a_{0}$ to a root $α \in Z_{p}$ of $f$ , and polynomial division produces a factorisation $f = (X - α) h$ with $h \in Z_{p} [X]$ reducing to $\overset{ˉ}{h}$ . Uniqueness of $α$ in the Hensel disc forces uniqueness of $h$ once the leading coefficients are normalised. Rubric: full credit for the reduction-to-Newton argument plus the polynomial-division factorisation.

Exercise 6 (medium, numeric).

Compute the four fourth roots of unity in $Z_{5}$ , i.e., the four elements $ζ \in Z_{5}$ with $ζ^{4} = 1$ , given to precision modulo $25$ .

Hint

The polynomial $X^{4} - 1$ factors over $F_{5}$ as $(X - 1) (X - 2) (X - 3) (X - 4) = (X - 1) (X - 4) (X^{2} + 1)$ , and each linear factor lifts via Hensel.

Answer

The four fourth roots of unity in $F_{5}$ are $1, 2, 3, 4$ (since $F_{5}^{\times}$ is cyclic of order $4$ ). Each is a simple root of $X^{4} - 1$ and lifts via Hensel:

$ζ = 1$ lifts to $1 \in Z_{5}$ exactly.
$ζ = 4 \equiv - 1$ lifts to $- 1 \in Z_{5}$ exactly.
$ζ \equiv 2 (mod 5)$ lifts to $a_{1} \equiv 7 (mod 25)$ (this is the Hensel-lifted square root of $- 1$ from the Beginner-tier worked example).
$ζ \equiv 3 (mod 5)$ lifts to $a_{1} \equiv 18 (mod 25)$ (the additive inverse of $7$ modulo $25$ ).

Together ${1, 7, 18, 24} (mod 25)$ are the four fourth roots of unity in $Z_{5} /25 Z_{5}$ , lifting to four $5$ -adic units forming the cyclic group $μ_{4} (Z_{5}) ≅ Z /4 Z$ . Rubric: full credit for the four residues with the Hensel-lift identification.

Exercise 7 (hard, short-answer).

Prove the Teichmüller-lift structure theorem: $Z_{p}^{\times} ≅ μ_{p - 1} \times (1 + p Z_{p})$ for odd primes $p$ , where $μ_{p - 1}$ is the cyclic group of $(p - 1)$ -th roots of unity.

Hint

Use Exercise 3 to produce $μ_{p - 1}$ inside $Z_{p}^{\times}$ . The reduction map $Z_{p}^{\times} \to F_{p}^{\times}$ has kernel $1 + p Z_{p}$ , and $μ_{p - 1}$ provides a section.

Answer

The reduction map $ρ : Z_{p}^{\times} \to F_{p}^{\times}$ has kernel $1 + p Z_{p}$ (the units congruent to $1$ modulo $p$ ), giving the short exact sequence $$ 1 \to 1 + p \mathbb{Z}_p \to \mathbb{Z}_p^\times \to \mathbb{F}p^\times \to 1. $$ By Exercise 3, Hensel's lemma applied to $X^{p - 1} - 1$ produces a subgroup $\mu{p-1} \subset \mathbb{Z}_p^\times $o f or d er$ p - 1 $ma pp in g i so m or p hi c a l l y o n t o$ \mathbb{F}_p^\times $. T h e * * T e i c hm \overset{u}{¨} l l er c ha r a c t er * *$ \omega : \mathbb{F}p^\times \to \mu{p-1} $i s t h e in v er seo f t hi sr es t r i c t i o n, an d i t i s a sec t i o n o f$ \rho $. T h ese q u e n ces pl i t s, so$ \mathbb{Z}p^\times \cong \mu{p-1} \times (1 + p \mathbb{Z}_p) $. T h e f a c t or$ 1 + p \mathbb{Z}_p $i s a p r o -$ p $g r o u p i so m or p hi c v ia t h e$ p $- a d i c l o g a r i t hm t o$ p \mathbb{Z}_p $(a dd i t i v e l y), h e n ce p r o -$ p $- cy c l i c f or$ p $o dd . P u tt in g t o g e t h er :$ \mathbb{Z}_p^\times \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p $a s t o p o l o g i c a l ab e l ian g r o u p s f or$ p $o dd . R u b r i c : f u l l cr e d i t f or t h es h or t - e x a c t - se q u e n ces pl i t v ia T e i c hm \overset{u}{¨} l l er pl u s t h e$ 1 + p\mathbb{Z}_p \cong \mathbb{Z}_p$ identification.

Exercise 8 (hard, short-answer).

State Krasner's lemma and use it to show that the algebraic closure $\overline{Q_{p}}$ is dense in its completion $C_{p}$ .

Hint

Krasner's lemma says: if $α, β$ are algebraic over $Q_{p}$ and $∣ β - α ∣_{p} < ∣ α^{'} - α ∣_{p}$ for every Galois conjugate $α^{'} \neq = α$ , then $Q_{p} (α) \subseteq Q_{p} (β)$ .

Answer

Krasner's lemma. Let $α \in \overline{Q_{p}}$ have minimal polynomial $f_{α} \in Q_{p} [X]$ with distinct Galois conjugates $α = α_{1}, α_{2}, \dots, α_{d}$ . If $β \in \overline{Q_{p}}$ satisfies $∣ β - α ∣_{p} < min_{i \geq 2} ∣ α - α_{i} ∣_{p}$ , then every Galois conjugate $σ (α)$ of $α$ (over $Q_{p} (β)$ ) is itself within the disc of radius $∣ β - α ∣_{p}$ around $α$ , forcing $σ (α) = α$ for every $σ$ fixing $β$ . Hence $α \in Q_{p} (β)$ and $Q_{p} (α) \subseteq Q_{p} (β)$ .

Application: density of $\overline{Q_{p}}$ in $C_{p}$ . Let $ξ \in C_{p}$ , the completion of $\overline{Q_{p}}$ , and let $ε > 0$ . Choose a Cauchy sequence $(β_{n}) \subset \overline{Q_{p}}$ converging to $ξ$ ; some $β_{n}$ lies within $ε$ of $ξ$ . So $\overline{Q_{p}}$ is dense in $C_{p}$ by construction, and Krasner's lemma is what guarantees the algebraic closure is itself algebraically closed of $Q_{p}$ after completion: any root of a polynomial over $C_{p}$ can be approximated by roots of nearby polynomials over $\overline{Q_{p}}$ , which by Krasner lie in finite extensions of $Q_{p}$ . The completion $C_{p}$ is therefore the algebraic closure of $Q_{p}$ after completion, and it is algebraically closed. Rubric: full credit for the lemma statement plus the density-and-algebraically-closed argument.

Advanced results Master

Theorem (Hensel's lemma, factorisation-lifting form; Bourbaki Commutative Algebra III §4 No. 3). Let $f \in Z_{p} [X]$ and suppose $\overset{ˉ}{f} \in F_{p} [X]$ factors as $\overset{ˉ}{f} = \overset{g}{ˉ}_{0} \overset{ˉ}{h}_{0}$ with $\overset{g}{ˉ}_{0}, \overset{ˉ}{h}_{0}$ coprime in $F_{p} [X]$ and $\overset{g}{ˉ}_{0}$ monic. Then there exist unique $g, h \in Z_{p} [X]$ with $f = g h$ , $g$ monic, $\overset{g}{ˉ} = \overset{g}{ˉ}_{0}$ , $\overset{ˉ}{h} = \overset{ˉ}{h}_{0}$ , and $de g g = de g \overset{g}{ˉ}_{0}$ .

The proof iterates a Bezout step: at each stage one has $f \equiv g_{n} h_{n} (mod p^{n + 1})$ with $\overset{g}{ˉ}_{n} = \overset{g}{ˉ}_{0}$ , $\overset{ˉ}{h}_{n} = \overset{ˉ}{h}_{0}$ , and uses coprimality of $\overset{g}{ˉ}_{0}, \overset{ˉ}{h}_{0}$ (yielding $u, v \in F_{p} [X]$ with $u \overset{g}{ˉ}_{0} + v \overset{ˉ}{h}_{0} = 1$ ) to construct correction terms $δ g_{n}, δ h_{n}$ with $g_{n + 1} = g_{n} + p^{n + 1} δ g_{n}$ and $h_{n + 1} = h_{n} + p^{n + 1} δ h_{n}$ satisfying $f \equiv g_{n + 1} h_{n + 1} (mod p^{n + 2})$ . The sequences $(g_{n}), (h_{n})$ converge in $Z_{p} [X]$ to the desired lift. Uniqueness is the same coprimality-and-degree argument as in the Newton-iteration form.

Theorem (Henselian valuation rings; Bosch-Güntzer-Remmert §3.4 Theorem 5). For a valued field $(K, ∣ \cdot ∣)$ with valuation ring $O_{K}$ and residue field $k$ , the following are equivalent:

$K$ is Henselian: every monic $f \in O_{K} [X]$ with a coprime residual factorisation lifts to a coprime factorisation in $O_{K} [X]$ .
Every simple residual root lifts: for every monic $f \in O_{K} [X]$ and every simple root $\overset{α}{ˉ} \in k$ of $\overset{ˉ}{f}$ , there exists a unique $α \in O_{K}$ with $f (α) = 0$ and $\overset{α}{ˉ}$ the reduction of $α$ .
The valuation extends uniquely to every algebraic extension of $K$ .
Every monic irreducible $f \in O_{K} [X]$ has $\overset{ˉ}{f} = (\overset{a}{ˉ} + b X^{n})^{m}$ for some $\overset{a}{ˉ}, b$ and $m, n$ depending on $f$ (the Newton-polygon characterisation).

Completeness of $K$ in the topology induced by $∣ \cdot ∣$ implies Henselian, but the converse fails: there exist Henselian non-complete valued fields (the Henselisation of any valued field is the minimal Henselian extension and is generally a proper subfield of the completion). The completion of a Henselian field is Henselian.

Theorem (Krasner's lemma; Krasner 1944). Let $K$ be a complete non-archimedean valued field of characteristic zero, let $α \in \overline{K}$ have distinct Galois conjugates $α = α_{1}, \dots, α_{d}$ , and let $β \in \overline{K}$ satisfy $$ |\beta - \alpha|K < \min{i \geq 2} |\alpha - \alpha_i|_K. $$ Then $K (α) \subseteq K (β)$ .

The proof: every Galois automorphism $σ$ of $\overline{K}$ over $K (β)$ permutes the $α_{i}$ and fixes $β$ , so $∣ σ (α) - α ∣_{K} = ∣ σ (α) - β + β - α ∣_{K} \leq max (∣ σ (α) - β ∣_{K}, ∣ β - α ∣_{K})$ . The first term equals $∣ σ (α - β) ∣_{K} = ∣ α - β ∣_{K}$ by isometry of the unique extension of the valuation, and the second equals $∣ β - α ∣_{K}$ ; both are strictly less than $min_{i \geq 2} ∣ α - α_{i} ∣_{K}$ . So $σ (α)$ is the unique $α_{i}$ within this distance of $α$ , namely $α_{1} = α$ . Thus every $σ$ fixing $β$ fixes $α$ , and Galois theory gives $K (α) \subseteq K (β)$ .

Theorem (extensions of $Q_{p}$ are parameterised by Eisenstein polynomials; Serre Ch. II §3 + Neukirch II §6). Every totally ramified extension $L / Q_{p}$ of degree $n$ is generated by a root $π \in L$ of an Eisenstein polynomial $f (X) = X^{n} + a_{n - 1} X^{n - 1} + \dots + a_{1} X + a_{0} \in Z_{p} [X]$ with $a_{i} \in p Z_{p}$ for all $i$ and $a_{0} \in / p^{2} Z_{p}$ . Conversely, every Eisenstein polynomial is irreducible and its root generates a totally ramified extension of degree $n$ .

The Newton-polygon analysis of an Eisenstein polynomial has slope $1/ n$ , so all roots have valuation $1/ n$ , generating a totally ramified extension of degree $n$ . Conversely, in any totally ramified extension $L / Q_{p}$ of degree $n$ , a uniformiser $π$ of $O_{L}$ has minimal polynomial $f (X) = \prod_{i} (X - σ_{i} π) = X^{n} + a_{n - 1} X^{n - 1} + \dots + a_{0}$ with each $a_{i} = (- 1)^{n - i} e_{i} (σ_{j} π)$ the $(n - i)$ -th elementary symmetric polynomial in the conjugates; comparison of valuations gives the Eisenstein form. Combined with Krasner's lemma, this characterises finite extensions of $Q_{p}$ as: an unramified part (corresponding to a finite extension of $F_{p}$ via Hensel-lifting of irreducible polynomials over $F_{p}$ ) followed by a totally ramified part (an Eisenstein polynomial).

Theorem (continuity of roots; Krasner's lemma corollary). Fix a monic polynomial $f \in Q_{p} [X]$ with distinct roots $α_{1}, \dots, α_{d} \in \overline{Q_{p}}$ . For every $ε > 0$ there exists $δ > 0$ such that every monic $g \in Q_{p} [X]$ of the same degree with $∣ f - g ∣_{\infty} < δ$ (coefficient-wise) has roots $β_{1}, \dots, β_{d} \in \overline{Q_{p}}$ within $ε$ of the $α_{i}$ after permutation, and $Q_{p} (α_{i}) = Q_{p} (β_{i})$ for the matched pair.

The first statement is continuity of roots in characteristic zero (which holds in any valued field where the implicit function theorem applies); the second is the structural consequence of Krasner: a sufficiently close approximation $β_{i}$ generates the same field extension as $α_{i}$ . Together they imply that the lattice of finite extensions of $Q_{p}$ inside a fixed algebraic closure is discrete in the topology induced by coefficient-norm of minimal polynomials, a finiteness statement that powers the classification of local fields.

Theorem (completeness via Hensel; Eisenbud §7.4 Theorem 7.3). For a Noetherian local ring $(R, m)$ , completeness of $R$ with respect to the $m$ -adic topology implies $R$ is Henselian. The converse is false: there exist Henselian Noetherian local rings that are not complete (e.g., the Henselisation of any Noetherian local ring at its maximal ideal).

This identifies Hensel's lemma as a structural property that is weaker than completeness but still captures the key lifting capability. In the abstract framework of étale-local algebra (Grothendieck's SGA), Henselian local rings are the natural objects: they have the étale-lifting property without requiring the full machinery of completion.

Theorem (Hensel's lemma as a fixed-point statement; Cassels Ch. 4). In the setting of the Newton-iteration form, the Newton map $T : Z_{p} \to Z_{p}$ , $T (a) = a - f (a) / f^{'} (a)$ , restricted to the disc ${a : ∣ a - a_{0} ∣_{p} \leq ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}}$ , is a contraction of contraction ratio $c < 1$ . The Banach fixed-point theorem (in the ultrametric setting) produces a unique fixed point, which is the unique root of $f$ in the disc.

The contraction is in the $p$ -adic absolute value: if $a, a^{'} \in Z_{p}$ lie in the disc, then $∣ T (a) - T (a^{'}) ∣_{p} \leq c ∣ a - a^{'} ∣_{p}$ with $c = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}^{2}$ . The contraction-mapping theorem in a complete ultrametric space produces the fixed point. This reframing identifies the abstract content of Hensel's lemma with the abstract content of Newton's method in real analysis: both are contraction-mapping theorems, and the difference between archimedean and non-archimedean settings is exactly the difference between iterated $\frac{1}{2}$ -contractions on $R$ (which converge geometrically) and ultrametric contractions on $Z_{p}$ (which converge quadratically with the bonus of automatic Cauchy structure).

Synthesis. Hensel's lemma is the foundational reason the $p$ -adic integers behave so differently from the ordinary integers: a residual approximation to a root always lifts, the Newton iteration converges quadratically, and the limit is unique in the Hensel disc. The central insight is that completeness of $Z_{p}$ in its ultrametric topology, combined with the strict inequality $∣ f (a_{0}) ∣_{p} < ∣ f^{'} (a_{0}) ∣_{p}^{2}$ , is exactly what makes Newton iteration into a contraction mapping with geometrically shrinking gap. Putting these together, Hensel's lemma identifies the existence of $p$ -adic roots with a finite computation in $F_{p}$ : check the residual root and its derivative; the rest is automatic. The bridge is the recognition that this structural property generalises to arbitrary complete discrete valuation rings, then to Henselian local rings without completeness, then to the abstract étale-local algebra of Grothendieck where Hensel becomes the structural input to the theory of étale morphisms.

The lemma also identifies several pairings between local algebra and Galois theory. Krasner's lemma is dual to Hensel in a precise sense: where Hensel lifts a root from $F_{p}$ to $Z_{p}$ , Krasner identifies which extensions of $Q_{p}$ are determined by approximate data. Together they produce the classification of finite extensions of $Q_{p}$ as Eisenstein-times-unramified, with the unramified part captured by Hensel-lifting of polynomials over $F_{p}$ and the ramified part by Eisenstein polynomials in $Z_{p} [X]$ . This is exactly the foundational reason that the Galois group $Gal (\overline{Q_{p}} / Q_{p})$ has the structure it does: an unramified quotient isomorphic to $\hat{Z} = Gal (\overline{F_{p}} / F_{p})$ on top of the wild-inertia normal subgroup. The lemma builds toward the local class field theory of 21.07.01, the $p$ -adic Galois representations of 21.05.01, and the formalism of local-global compatibility in the Langlands programme of 21.10.01. Every place where the local information at a prime $p$ is leveraged to produce a global arithmetic conclusion, Hensel's lemma is the constructive input that makes the local information actionable.

Full proof set Master

Theorem (Hensel's lemma, Newton-iteration form), proof. Given in the Intermediate-tier section: define the Newton iteration $a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})$ , check via Taylor expansion that each step preserves $a_{n} \in Z_{p}$ and that $∣ f (a_{n + 1}) ∣_{p} \leq c ∣ f (a_{n}) ∣_{p}$ with $c = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}^{2} < 1$ . The geometric bound $∣ f (a_{n}) ∣_{p} \leq c^{2^{n} - 1} ∣ f (a_{0}) ∣_{p}$ (quadratic convergence) plus completeness of $Z_{p}$ produces the Cauchy limit, and ultrametric uniqueness in the Hensel disc closes the argument. $□$

Theorem (factorisation-lifting form), proof. Set $g_{0}, h_{0} \in Z_{p} [X]$ as monic lifts of $\overset{g}{ˉ}_{0}, \overset{ˉ}{h}_{0}$ with the same degrees, normalised so $de g g_{0} = de g \overset{g}{ˉ}_{0}$ and $f \equiv g_{0} h_{0} (mod p)$ . Inductively suppose $g_{n}, h_{n} \in Z_{p} [X]$ satisfy $f \equiv g_{n} h_{n} (mod p^{n + 1})$ with the same reductions and degrees. Write $f - g_{n} h_{n} = p^{n + 1} R_{n}$ with $R_{n} \in Z_{p} [X]$ . Coprimality of $\overset{g}{ˉ}_{0}, \overset{ˉ}{h}_{0}$ in $F_{p} [X]$ produces $\overset{u}{ˉ}_{0}, \overset{v}{ˉ}_{0} \in F_{p} [X]$ with $\overset{u}{ˉ}_{0} \overset{g}{ˉ}_{0} + \overset{v}{ˉ}_{0} \overset{ˉ}{h}_{0} = 1$ , and we can choose $\overset{v}{ˉ}_{0}$ to have degree less than $de g \overset{g}{ˉ}_{0}$ . Lift $\overset{u}{ˉ}_{0}, \overset{v}{ˉ}_{0}$ to $u_{0}, v_{0} \in Z_{p} [X]$ with $u_{0} g_{0} + v_{0} h_{0} \equiv 1 (mod p)$ .

Define $δ h_{n} = \overset{ˉ}{R}_{n} \cdot \overset{u}{ˉ}_{0}$ reduced modulo $\overset{ˉ}{h}_{0}$ (giving $de g δ h_{n} < de g \overset{ˉ}{h}_{0}$ ), and $δ g_{n} = \overset{ˉ}{R}_{n} \cdot \overset{v}{ˉ}_{0}$ reduced modulo $\overset{g}{ˉ}_{0}$ , with appropriate adjustment to maintain $de g g_{n + 1} = de g \overset{g}{ˉ}_{0}$ . Set $g_{n + 1} = g_{n} + p^{n + 1} δ g_{n}$ and $h_{n + 1} = h_{n} + p^{n + 1} δ h_{n}$ . Then $g_{n + 1} h_{n + 1} = g_{n} h_{n} + p^{n + 1} (g_{n} δ h_{n} + δ g_{n} h_{n}) + p^{2 (n + 1)} δ g_{n} δ h_{n} \equiv g_{n} h_{n} + p^{n + 1} (g_{0} δ h_{n} + δ g_{n} h_{0}) (mod p^{n + 2})$ , and the bracket equals $R_{n}$ by choice of $δ g_{n}, δ h_{n}$ . So $f \equiv g_{n + 1} h_{n + 1} (mod p^{n + 2})$ , completing the induction.

The sequences $(g_{n}), (h_{n})$ are Cauchy in $Z_{p} [X]$ coefficient-wise (each new term differs from the previous by a polynomial with coefficients in $p^{n + 1} Z_{p}$ ), so they converge to $g, h \in Z_{p} [X]$ with $f = g h$ , $\overset{g}{ˉ} = \overset{g}{ˉ}_{0}$ , $\overset{ˉ}{h} = \overset{ˉ}{h}_{0}$ , and the degree constraint. Uniqueness: suppose $g^{'}, h^{'} \in Z_{p} [X]$ is another lift with the same data. The difference $g - g^{'}$ vanishes modulo $p$ (since both reduce to $\overset{g}{ˉ}_{0}$ ), and the same for $h - h^{'}$ . The product condition $g h = g^{'} h^{'} = f$ plus coprimality of reductions forces $g = g^{'}$ and $h = h^{'}$ by an ultrametric Bezout argument. $□$

Theorem (Henselian valuation rings, equivalences), proof sketch. The implications $(1) ⟹ (2)$ is the special case of factorisation lifting where one factor is linear. $(2) ⟹ (3)$ : a non-unique extension of the valuation would produce a non-unique factorisation of some minimal polynomial, contradicting unique root lifting. $(3) ⟹ (4)$ : a polynomial with two distinct factor-types in the residue field would force two distinct valuation extensions on its splitting field. $(4) ⟹ (1)$ : the Newton-polygon characterisation determines the irreducible factors of $f$ once the residual factorisation is given, and the Newton-iteration form of Hensel produces each factor as a $K$ -rational polynomial. Full details are in Bosch-Güntzer-Remmert §3.4. $□$

Theorem (Krasner's lemma), proof. Let $σ$ be a $K (β)$ -automorphism of $\overline{K}$ . Since $σ$ fixes $β$ , we have $∣ σ (α) - β ∣_{K} = ∣ σ (α - β) ∣_{K} = ∣ α - β ∣_{K}$ (the valuation is preserved by $K$ -isometries of $\overline{K}$ , in particular by $σ$ ). The ultrametric inequality gives $$ |\sigma(\alpha) - \alpha|_K \leq \max(|\sigma(\alpha) - \beta|_K, |\beta - \alpha|_K) = |\beta - \alpha|K < \min{i \geq 2} |\alpha - \alpha_i|_K. $$ But $σ (α)$ is one of the conjugates $α_{1}, \dots, α_{d}$ , and the inequality just established rules out every $α_{i}$ with $i \geq 2$ . So $σ (α) = α_{1} = α$ , meaning every $σ$ fixing $β$ also fixes $α$ . Galois theory gives $α \in K (β)$ and hence $K (α) \subseteq K (β)$ . $□$

Theorem (Eisenstein characterisation of totally ramified extensions), proof sketch. Forward: let $L / Q_{p}$ be totally ramified of degree $n$ . The ring $O_{L}$ is a DVR with maximal ideal $m_{L} = (π)$ for a uniformiser $π$ , with $∣ π ∣_{L} = p^{- 1/ n}$ . The minimal polynomial $f$ of $π$ over $Q_{p}$ has degree $n$ (since $π$ generates the extension), and its coefficients are elementary symmetric polynomials in the Galois conjugates $π_{1}, \dots, π_{n}$ , all of valuation $1/ n$ . The valuation of the $i$ -th coefficient is at least $(n - i) / n$ rounded up to the nearest integer, hence at least $1$ for $i < n$ . The constant term has valuation $1$ , not more (since the product of conjugates is $π \cdot \prod_{i \geq 2} π_{i}$ with each factor of valuation $1/ n$ , total valuation $1$ ). This is the Eisenstein condition.

Reverse: Eisenstein polynomial $f$ has all coefficients except the leading one in $p Z_{p}$ , with constant term not in $p^{2} Z_{p}$ . Newton-polygon analysis: the slopes of the Newton polygon of $f$ are determined by the valuations of the coefficients, and the Eisenstein condition forces a single slope of $1/ n$ . So all roots have valuation $1/ n$ , and the field they generate is totally ramified of degree $n$ . Irreducibility: a factorisation $f = g h$ with $g, h \in Z_{p} [X]$ monic of degrees $a, b > 0$ would force one of the factors to have a constant term of valuation $0$ (the slopes would split into pieces of total slope $1$ ), contradicting that the constant term of $f$ has valuation exactly $1$ . $□$

Theorem (continuity of roots and Krasner), proof. Continuity of roots in $C$ holds for polynomials with simple roots: if $f, g$ have the same degree and $g$ is sufficiently close to $f$ coefficient-wise, the roots of $g$ are close to the roots of $f$ in the metric. The same argument works in $\overline{Q_{p}}$ with the unique extension of the $p$ -adic absolute value. Once one chooses $δ$ small enough that the perturbation $∣ f - g ∣_{\infty} < δ$ produces roots $β_{i}$ within $ε$ of $α_{i}$ , Krasner's lemma applies: $β_{i}$ is within the disc of radius $min_{j \neq = i} ∣ α_{i} - α_{j} ∣_{K} /2$ of $α_{i}$ , so $Q_{p} (α_{i}) \subseteq Q_{p} (β_{i})$ . The same argument with the roles reversed (perturbing back from $g$ to $f$ ) gives $Q_{p} (β_{i}) \subseteq Q_{p} (α_{i})$ , hence equality. $□$

Theorem (completeness implies Henselian), proof. Suppose $(R, m)$ is a complete Noetherian local ring, and let $f \in R [X]$ be monic with a residual coprime factorisation $\overset{ˉ}{f} = \overset{g}{ˉ}_{0} \overset{ˉ}{h}_{0}$ in $(R / m) [X]$ . Choose monic lifts $g_{0}, h_{0} \in R [X]$ and run the same Bezout iteration as in the $Z_{p}$ proof, with $m^{n + 1}$ replacing $p^{n + 1}$ at each step. The Cauchy property of the iterates plus $m$ -adic completeness of $R$ produces the lift. Non-completeness allows non-Henselian counterexamples: the localisation $Z_{(p)}$ at the prime $(p)$ is a Noetherian local ring, not complete and not Henselian; its Henselisation $Z_{(p)}^{h}$ is a proper subring of $Z_{p}$ . $□$

Theorem (Hensel as a fixed-point statement), proof. The Newton map $T (a) = a - f (a) / f^{'} (a)$ restricted to the Hensel disc $D = {a \in Z_{p} : ∣ a - a_{0} ∣_{p} \leq r}$ with $r = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}$ has the property $$ T(a) - T(b) = (a - b) - \big(f(a) - f(b)\big)/f'(a) + f(b)(1/f'(a) - 1/f'(b)). $$ Using $f (a) - f (b) = (a - b) (f^{'} (a) + (a - b) S (a, b))$ for some $S$ in two variables with $S$ bounded by Taylor remainder, and the constancy of $∣ f^{'} (a) ∣_{p} = ∣ f^{'} (a_{0}) ∣_{p}$ on $D$ from Step 1 of the Newton-iteration proof, one estimates $∣ T (a) - T (b) ∣_{p} \leq c ∣ a - b ∣_{p}$ with $c = ∣ f (a_{0}) ∣_{p} /∣ f^{'} (a_{0}) ∣_{p}^{2} < 1$ . The ultrametric Banach fixed-point theorem on the complete metric space $D$ gives a unique fixed point, which is the unique root of $f$ in $D$ . $□$

Connections Master

Absolute value and triangle inequality 00.01.02. The $p$ -adic absolute value is the ultrametric refinement of the ordinary absolute value introduced in this foundational unit. The ultrametric inequality $∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ is the load-bearing analytic input to Hensel's lemma: it is what makes the Taylor-remainder estimate $∣ ε^{2} R (ε) ∣_{p} \leq ∣ ε ∣_{p}^{2}$ strictly smaller than the linear term $∣ ε f^{'} (a) ∣_{p} = ∣ ε ∣_{p} \cdot ∣ f^{'} (a_{0}) ∣_{p}$ , and hence what powers the quadratic convergence of the Newton iteration.
Cauchy sequences and completeness 02.03.02. Hensel's lemma constructs a Cauchy sequence of approximations and produces the root as the limit. Completeness of $Z_{p}$ in its ultrametric topology is the essential input: an analogous Newton iteration over $Z$ would produce a Cauchy sequence with no integer limit, and the construction collapses. The completion of $Z$ at the $p$ -adic absolute value is exactly what makes Hensel's lemma actionable.
Algebraic field extensions 01.02.12. Krasner's lemma, the natural companion to Hensel, characterises finite extensions of $Q_{p}$ . Combined with the Eisenstein characterisation of totally ramified extensions, the result is the classification of all finite extensions of $Q_{p}$ as an unramified part (lifted from extensions of $F_{p}$ via Hensel) followed by a totally ramified part (an Eisenstein polynomial). This is the local analogue of the global theory of algebraic number fields.
$Z_{p}$ -extensions and Iwasawa theory 21.07.01. The Iwasawa algebra $Λ = Z_{p} [[T]]$ is the foundational object of Iwasawa theory, and its analytic structure rests on Hensel-type lifting arguments: roots of polynomials over $Λ$ are constructed by analogous Newton iterations, and the Weierstrass preparation theorem (which factors any non-zero $f \in Λ$ as $p^{μ} \cdot u (T) \cdot P (T)$ with $P$ distinguished) is the $Λ$ -coefficient analogue of the factorisation-lifting form of Hensel.
$ℓ$ -adic Galois representations 21.05.01. The decomposition group $D_{p} \subset Gal (\overline{Q} / Q)$ at a prime $p$ is naturally identified with $Gal (\overline{Q_{p}} / Q_{p})$ , whose structure is exactly the unramified-times-wildly-ramified picture made possible by Hensel and Krasner. The inertia subgroup $I_{p}$ , the wild inertia $P_{p}$ , and the tame quotient $I_{p} / P_{p}$ all rest on the local-field machinery whose constructive input is Hensel's lemma.

Historical & philosophical context Master

Kurt Hensel introduced the $p$ -adic numbers in 1897 in a short note in the Jahresbericht der Deutschen Mathematiker-Vereinigung ^{[Hensel 1897]}, motivated by an analogy with the power-series construction in algebraic geometry: just as the local ring of a smooth point on a curve is a power-series ring $k [[t]]$ in one variable, the local ring at a prime $p$ of $Z$ should be the completion $Z_{p}$ . The factorisation lifting that bears his name was the constructive input that made the analogy work: a polynomial over $Z_{p}$ should factor in step with its reduction modulo $p$ , just as a function on a smooth curve should factor in step with its restriction to a fibre. Hensel published the systematic treatment in his 1908 monograph Theorie der algebraischen Zahlen (Teubner, Leipzig) ^{[Hensel 1908]}, where Hensel's lemma appears in essentially its modern form, with proofs by direct computation rather than the Newton-iteration packaging that became standard later.

The Newton-iteration framing arose from work in the 1920s and 1930s on $p$ -adic analysis. Helmut Hasse used Hensel's lemma centrally in his 1923 doctoral thesis under Hensel and in his subsequent program identifying the local-global principle for quadratic forms (the Hasse-Minkowski theorem: a quadratic form over $Q$ has a non-zero rational zero if and only if it has a non-zero zero in every $Q_{p}$ and in $R$ ). Marc Krasner published his eponymous lemma in 1944 in the Journal de Mathématiques Pures et Appliquées ^{[Krasner 1944]} (with a preliminary version in Mathematica Cluj 1937), giving the structural complement to Hensel: where Hensel lifts a root, Krasner identifies which approximate roots already determine the field they generate. Together Hensel and Krasner produce the full classification of finite extensions of $Q_{p}$ . The completion of this classification was carried out in the post-war years, with Serre's Corps Locaux (1962) and Cassels-Fröhlich's Algebraic Number Theory (1967) establishing the modern presentation.

Alexander Grothendieck's generalisation to commutative algebra came in the 1950s and 1960s. In the Éléments de géométrie algébrique and its appendices, Hensel's lemma is abstracted to the definition of a Henselian local ring: a local ring where every simple residual root lifts. The Henselisation construction (the minimal Henselian extension of a given local ring) became the bridge between the analytic theory of $p$ -adic numbers and the algebraic theory of étale morphisms, and Hensel's lemma was identified as the defining feature of "étale-local" algebra. The 1971 SGA seminar exposés on étale fundamental groups exploit this connection systematically, and the modern statement of Hensel's lemma — that a complete local ring is Henselian — is now standard.

Bibliography Master

@article{Hensel1897,
  author  = {Hensel, Kurt},
  title   = {{\"U}ber eine neue Begr{\"u}ndung der Theorie der algebraischen Zahlen},
  journal = {Jahresbericht der Deutschen Mathematiker-Vereinigung},
  volume  = {6},
  year    = {1897},
  pages   = {83--88}
}

@book{Hensel1908,
  author    = {Hensel, Kurt},
  title     = {Theorie der algebraischen Zahlen},
  publisher = {Teubner},
  address   = {Leipzig},
  year      = {1908}
}

@article{Krasner1944,
  author  = {Krasner, Marc},
  title   = {Sur la primitivit{\'e} des corps $p$-adiques},
  journal = {J. Math. Pures Appl.},
  volume  = {23},
  year    = {1944},
  pages   = {67--92}
}

@book{Serre1973,
  author    = {Serre, Jean-Pierre},
  title     = {A Course in Arithmetic},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {7},
  year      = {1973}
}

@book{SerreCorpsLocaux,
  author    = {Serre, Jean-Pierre},
  title     = {Corps Locaux},
  publisher = {Hermann},
  address   = {Paris},
  year      = {1962}
}

@book{Cassels1986LocalFields,
  author    = {Cassels, J. W. S.},
  title     = {Local Fields},
  publisher = {Cambridge University Press},
  series    = {London Mathematical Society Student Texts},
  volume    = {3},
  year      = {1986}
}

@book{Neukirch1999,
  author    = {Neukirch, J{\"u}rgen},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {322},
  year      = {1999}
}

@book{Koblitz1984,
  author    = {Koblitz, Neal},
  title     = {$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {58},
  edition   = {2},
  year      = {1984}
}

@book{Gouvea1997,
  author    = {Gouv{\^e}a, Fernando Q.},
  title     = {$p$-adic Numbers: An Introduction},
  publisher = {Springer},
  series    = {Universitext},
  edition   = {2},
  year      = {1997}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yu. I. and Panchishkin, A. A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2},
  year      = {2005}
}

@book{Eisenbud1995,
  author    = {Eisenbud, David},
  title     = {Commutative Algebra with a View toward Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {150},
  year      = {1995}
}

@book{BoschGuntzerRemmert1984,
  author    = {Bosch, Siegfried and G{\"u}ntzer, Ulrich and Remmert, Reinhold},
  title     = {Non-Archimedean Analysis},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {261},
  year      = {1984}
}

@article{Hasse1923,
  author  = {Hasse, Helmut},
  title   = {{\"U}ber die Darstellbarkeit von Zahlen durch quadratische Formen im K{\"o}rper der rationalen Zahlen},
  journal = {J. reine angew. Math.},
  volume  = {152},
  year    = {1923},
  pages   = {129--148}
}

Prerequisites

00.01.01
00.01.02
02.03.02
01.01.01
01.02.12

Tier anchors

beginner: Gouvêa *p-adic Numbers: An Introduction* (Springer Universitext, 2nd ed. 1997) Ch. 3 informal opening on lifting solutions modulo $p^n$; Koblitz *p-adic Numbers, p-adic Analysis, and Zeta-Functions* (Springer GTM 58, 2nd ed. 1984) Ch. I §6 introduction
intermediate: Serre *A Course in Arithmetic* (Springer GTM 7, 1973) Ch. II §2 (Hensel's lemma, Newton iteration, applications); Cassels *Local Fields* (LMS Student Texts 3, 1986) Ch. 4 §4; Neukirch *Algebraic Number Theory* (Springer Grundlehren 322, 1999) Ch. II §4
master: Hensel 1897 *Jahresber. Deutsch. Math.-Verein.* 6, 83-88 (originator: factorisation lifting from $\mathbb{F}_p$ to $\mathbb{Z}_p$); Hensel 1908 *Theorie der algebraischen Zahlen* (Teubner, Leipzig — first systematic treatment); Krasner 1944 *J. Math. Pures Appl.* 23 (Krasner's lemma and the structure of extensions of $\mathbb{Q}_p$); Serre 1973 Ch. II §2; Cassels 1986 Ch. 4; Neukirch 1999 Ch. II §§4-6; Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 1 §3; Bosch-Güntzer-Remmert *Non-Archimedean Analysis* (Springer Grundlehren 261, 1984) §3.4 (Henselian valuation rings); Eisenbud *Commutative Algebra with a View toward Algebraic Geometry* (Springer GTM 150, 1995) §7.4 (Hensel's lemma in commutative-algebra framing)

References

Hensel, K. — Über eine neue Begründung der Theorie der algebraischen Zahlen · *Jahresbericht der Deutschen Mathematiker-Vereinigung* 6 (1897), 83-88. The originator paper introducing $p$-adic numbers and the factorisation lifting that bears his name.
Hensel, K. — Theorie der algebraischen Zahlen · Teubner, Leipzig (1908). The first systematic monograph treatment of $p$-adic numbers, $p$-adic analytic functions, and the lifting lemma applied to algebraic-number-theoretic problems.
Krasner, M. — Sur la primitivité des corps p-adiques · *Mathematica Cluj* 13 (1937), 72-191; refined in *J. Math. Pures Appl.* 23 (1944), 67-92. The originator paper for Krasner's lemma: if $\alpha, \beta$ are algebraic over $\mathbb{Q}_p$ with $|\beta - \alpha|_p < |\alpha' - \alpha|_p$ for every Galois conjugate $\alpha' \neq \alpha$ of $\alpha$, then $\mathbb{Q}_p(\alpha) \subseteq \mathbb{Q}_p(\beta)$.
Serre, J.-P. — A Course in Arithmetic · Springer GTM 7 (1973). Ch. II §2 statement of Hensel's lemma in the Newton-iteration form and its standard corollaries (Teichmüller lift, structure of $\mathbb{Z}_p^\times$, squares in $\mathbb{Q}_p^\times$); Ch. II §1 construction of $\mathbb{Z}_p$ as an inverse limit.
Cassels, J. W. S. — Local Fields · London Mathematical Society Student Texts 3 (1986). Ch. 4 §4 Hensel's lemma in the Newton-iteration form; Ch. 7 Krasner's lemma and the characterisation of finite extensions of $\mathbb{Q}_p$.
Neukirch, J. — Algebraic Number Theory · Springer Grundlehren 322 (1999). Ch. II §4 Hensel's lemma and Henselian valuation rings; §6 unramified, tamely ramified, and totally ramified extensions of $\mathbb{Q}_p$.
Koblitz, N. — p-adic Numbers, p-adic Analysis, and Zeta-Functions · Springer GTM 58, 2nd ed. (1984). Ch. I §6 Hensel's lemma with explicit computations; Ch. I §5 the $p$-adic absolute value and ultrametric inequality.
Gouvêa, F. Q. — p-adic Numbers: An Introduction · Springer Universitext, 2nd ed. (1997). Ch. 3 Hensel's lemma, worked examples computing roots of unity in $\mathbb{Z}_p$, the Teichmüller lift; Ch. 2 the $p$-adic absolute value and its completion.
Manin, Yu. I. & Panchishkin, A. A. — Introduction to Modern Number Theory · Springer Encyclopaedia of Mathematical Sciences 49, 2nd ed. (2005). Ch. 1 §3 the $p$-adic numbers, Hensel's lemma, and applications to Diophantine equations.
Eisenbud, D. — Commutative Algebra with a View toward Algebraic Geometry · Springer GTM 150 (1995). §7.4 the abstract form of Hensel's lemma for complete local rings, and the connection between Henselisation and étale-local algebra.
Bosch, S., Güntzer, U. & Remmert, R. — Non-Archimedean Analysis · Springer Grundlehren 261 (1984). §3.4 Henselian valuation rings and the formulation of Hensel's lemma as a defining property of completeness for non-archimedean valuations.

Estimated time

beginner: 18m
intermediate: 40m
master: 80m