Questions: Arithmetic in p-adic Numbers

Question 1

A polynomial f(x) has a root r₀ with f(r₀) ≡ 0 (mod p). Under what additional condition does Hensel's lemma guarantee that this root lifts to an exact root in ℚ_p?

• A. No additional condition is needed — any modular root can always be lifted
• B. f'(r₀) ≢ 0 (mod p), meaning the derivative at the root is not divisible by p
• C. The polynomial must have degree at most 2
• D. The prime p must be odd

Answer: (B)

Hensel's lemma requires a non-degeneracy condition: the derivative f'(r₀) must not be divisible by p. Without this, the root may be 'singular' modulo p and may not lift uniquely — or at all. Option A describes a common misconception: a modular root is necessary but not sufficient. The derivative condition ensures the Newton's-method iteration converges geometrically in the p-adic metric.

Question 2

Which philosophical principle does Hensel's lemma most directly support when combined with the Hasse principle for quadratic forms?

• A. The local-global philosophy: understanding a problem at every prime (and at ∞) gives information about global rational solutions
• B. The completeness of ℚ_p: every Cauchy sequence in the p-adic metric converges
• C. The ultrametric inequality: p-adic absolute values satisfy |x + y|_p ≤ max(|x|_p, |y|_p)
• D. The uniqueness of prime factorization in the integers

Answer: (A)

The Hasse principle (for quadratic forms) states that a quadratic equation has a rational solution if and only if it has a solution in ℝ and in ℚ_p for every prime p. Hensel's lemma supplies the p-adic solutions by lifting modular ones. Together they exemplify the local-global philosophy: checking everywhere locally (at each prime p and at infinity) suffices to answer the global question over ℚ.

Question 3

If a polynomial f(x) has a root modulo nearly every prime p, then it necessarily has a root in ℚ.

• T. True
• F. False

Answer: False

This is false in general — the Hasse principle does NOT hold for polynomials of degree higher than 2. For quadratic forms it is a theorem, but for higher-degree polynomials there can be 'failures of the Hasse principle': a polynomial may have roots in every ℚ_p (and in ℝ) yet have no rational root. The Hasse principle is a special property of quadratic forms, not a universal law.

Question 4

The p-adic integers ℤ_p are exactly the elements of ℚ_p with p-adic valuation greater than or equal to zero.

• T. True
• F. False

Answer: True

True. The p-adic valuation v_p(x) measures divisibility by p: elements with v_p(x) ≥ 0 are those not in the 'denominator' part of the p-adic expansion. These form the ring of integers ℤ_p inside the field ℚ_p, analogous to how ℤ sits inside ℚ. Hensel's lemma, when the root is in ℤ/pℤ, lifts it into ℤ_p.

Question 5

Explain in your own words what 'lifting' means in Hensel's lemma, and why the p-adic completion is the natural setting for it.

Model answer: Lifting means starting with an approximate solution (a root mod p) and systematically refining it into increasingly precise solutions (mod p², mod p³, ...) until you have an infinite sequence that converges to an exact root in ℚ_p. The p-adic completion is natural because the p-adic metric makes these approximations converge: each step multiplies the error by p, so errors shrink toward zero in the p-adic sense. The completed field ℚ_p guarantees that the infinite sequence of consistent approximations has an actual limit.

The key analogy is Newton's method in calculus: each iteration of the Hensel lift doubles the number of correct digits (in the p-adic expansion), just as Newton's method doubles the correct decimal places in a real root. The p-adic metric is what makes 'convergence' meaningful — in ordinary ℤ, the sequence of residues grows without bound, but in ℚ_p they converge because divisibility by higher powers of p means 'smaller.'