Questions: Hensel's Lemma

Question 1

Suppose f(x) = x² − 2 and you find that f(3) ≡ 0 (mod 7), with f'(3) = 6 ≢ 0 (mod 7). What does Hensel's Lemma guarantee?

• A. There exists at least one solution to f(x) ≡ 0 (mod 49), but it may not be unique
• B. There exists exactly one solution to f(x) ≡ 0 (mod 7^k) for every k ≥ 1, congruent to 3 mod 7
• C. The solution lifts to mod 49 but further lifting requires checking again whether the derivative remains nonzero
• D. There is no guarantee of a lift unless 2 divides p − 1

Answer: (B)

When f'(a) ≢ 0 (mod p), Hensel's Lemma guarantees a *unique* lift at every stage: from mod p to mod p², from mod p² to mod p³, and so on indefinitely. The nonzero derivative condition does not need to be re-verified at each stage — once it holds mod p, the lifting works all the way up, yielding a unique element of ℤ_p. Option A is too weak (uniqueness is guaranteed), and option C misunderstands the iterative structure.

Question 2

Suppose f(a) ≡ 0 (mod p) and f'(a) ≡ 0 (mod p). What does Hensel's Lemma say about lifting a to a solution mod p²?

• A. The lift is guaranteed to exist but may not be unique
• B. The lift is guaranteed to be unique but may not exist
• C. Hensel's Lemma gives no guarantee: there may be no lift, or multiple lifts
• D. The lift always exists because the original solution is valid mod p

Answer: (C)

The nonzero derivative condition is precisely what makes Hensel's Lemma work. When f'(a) ≡ 0 (mod p), the argument breaks down: the linear equation for the lift t has no solution (if f(a)/p ≢ 0 mod p there is no lift), or infinitely many solutions (if f(a)/p ≡ 0 mod p, any t works, giving p distinct lifts). This parallels Newton's method failing at a repeated real root. The lesson: the derivative condition is not a technicality but the essential mechanism of uniqueness.

Question 3

Hensel's Lemma applies to a polynomial f over ℤ, and we find a solution a mod p. Under the conditions of the lemma, this solution lifts to a unique solution in ℤ_p — the p-adic integers.

• T. True
• F. False

Answer: True

This is exactly the content of Hensel's Lemma. When f(a) ≡ 0 (mod p) and f'(a) ≢ 0 (mod p), the iterative lifting process gives a unique sequence a, a₁, a₂, ... where aₖ ≡ a (mod p) and f(aₖ) ≡ 0 (mod p^{k+1}). This sequence is coherent (aₖ ≡ a_{k-1} mod p^k) and thus defines a unique element of ℤ_p — the inverse limit of the system ℤ/p^k ℤ.

Question 4

If f'(a) ≡ 0 (mod p) at a solution a, then Hensel's Lemma still guarantees a lift exists, just without uniqueness.

• T. True
• F. False

Answer: False

When f'(a) ≡ 0 (mod p), Hensel's Lemma gives no guarantee at all — not existence, not uniqueness. The Taylor expansion mod p² gives f(a + tp) ≡ f(a) + tp·f'(a) ≡ f(a) (mod p²), so whether a lift exists depends entirely on whether f(a) ≡ 0 (mod p²). If f(a) ≢ 0 (mod p²), there is no lift; if f(a) ≡ 0 (mod p²), then any choice of t works (p lifts exist). The vanishing derivative is not just a uniqueness problem — it signals a complete failure of the Newton's-method mechanism.

Question 5

Why is the condition f'(a) ≢ 0 (mod p) the key to Hensel's Lemma, and what role does it play in the lifting argument?

Model answer: The nonzero derivative condition guarantees that the linear equation for the correction term t has a unique solution mod p. In the lifting step, writing b = a + tp and expanding f(b) ≡ f(a) + tp·f'(a) (mod p²), setting this to zero requires t ≡ −[f(a)/p]·[f'(a)]⁻¹ (mod p). For this to have a unique solution, two things must hold: f(a) must be divisible by p (the hypothesis), and f'(a) must be invertible mod p (the derivative condition). The derivative condition ensures the inverse exists, yielding a unique t and hence a unique lift.

This mirrors Newton's method for real roots: the tangent line at a near-root gives a unique improved estimate, provided the derivative is nonzero (so the tangent line isn't horizontal). A zero derivative would mean the tangent line is flat — it never crosses zero, or is identically zero, giving no information about where the root is. Hensel's Lemma is this same geometric intuition transported to the p-adic world.