Questions: p-adic Valuation

Question 1

A student calculates v₂(12) = 2 (since 12 = 2²·3) and v₂(1000) = 3 (since 1000 = 2³·125), then concludes that 12 is 2-adically closer to zero than 1000 because 12 is smaller in magnitude. Is this correct?

• A. Yes — the p-adic valuation preserves the ordinary ordering of integers
• B. No — higher valuation means larger p-adic absolute value, so both conclusions are wrong
• C. No — higher valuation means smaller p-adic absolute value, so 1000 is 2-adically closer to zero than 12
• D. It depends on which prime p is chosen

Answer: (C)

The p-adic absolute value is |x|_p = p^{−v_p(x)}. Since v₂(1000) = 3 > v₂(12) = 2, we get |1000|₂ = 2⁻³ = 1/8 < |12|₂ = 2⁻² = 1/4. So 1000 is 2-adically closer to zero than 12, despite being larger in ordinary magnitude. The p-adic world inverts intuition: 'large' numbers (that are highly divisible by p) are p-adically 'small.'

Question 2

The p-adic valuation v_p satisfies v_p(ab) = v_p(a) + v_p(b) for all nonzero rationals. This property makes the valuation analogous to which familiar function?

• A. A polynomial — it counts the degree of divisibility by p
• B. A logarithm — multiplication in the domain becomes addition in the range
• C. An exponential — the values grow rapidly with the number of prime factors
• D. A modular arithmetic function — values cycle through a fixed period

Answer: (B)

Just as log(ab) = log(a) + log(b), we have v_p(ab) = v_p(a) + v_p(b). Both functions convert multiplication to addition. The analogy goes further: v_p is 'log base p of the p-part of a number.' This additive structure is what makes the valuation so algebraically tractable and is what allows the p-adic absolute value to satisfy the ultrametric inequality.

Question 3

In the 5-adic absolute value, the integer 5,000,000 is 'closer to zero' than the integer 7.

• T. True
• F. False

Answer: True

5,000,000 = 5⁶ · 2⁶, so v₅(5,000,000) = 6 and |5,000,000|₅ = 5⁻⁶ ≈ 0.000064. For 7, v₅(7) = 0 and |7|₅ = 1. So |5,000,000|₅ << |7|₅, meaning 5,000,000 is far closer to zero in the 5-adic metric. This is the central conceptual reversal: p-adic 'smallness' measures divisibility by p, not magnitude on the number line.

Question 4

The p-adic absolute value satisfies primarily the ordinary triangle inequality |x + y|_p ≤ |x|_p + |y|_p, just like the usual absolute value.

• T. True
• F. False

Answer: False

The p-adic absolute value satisfies the strictly stronger ultrametric inequality: |x + y|_p ≤ max(|x|_p, |y|_p). Since max(a, b) ≤ a + b, the ultrametric inequality implies the ordinary triangle inequality, but not vice versa. The ultrametric inequality has unusual geometric consequences: every triangle in the p-adic metric is isoceles, and every point in an open ball is a center. This is what makes p-adic geometry so different from real analysis.

Question 5

Why does the p-adic valuation define a notion of 'size' in which large integers can be p-adically small?

Model answer: Because p-adic size measures how divisible a number is by p, not its magnitude. A number like 2^100 is 2-adically tiny (|2^100|₂ = 2^{-100}) even though it is astronomically large ordinarily. The valuation extracts one specific prime-exponent from the factorization and treats it as the sole measure of closeness to zero — an assignment orthogonal to ordinary magnitude.

The Fundamental Theorem of Arithmetic guarantees a unique factorization, and the p-adic valuation simply reads off one coordinate of that factorization. Different primes give different 'lenses' on the integers, each measuring a different kind of divisibility. None of them corresponds to ordinary magnitude, which is why p-adic geometry is genuinely different geometry, not just a rescaling of the familiar number line.