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p-adic Numbers — Definition, Formula & Examples

p-adic numbers are a system of numbers built from a chosen prime pp, where "closeness" is determined by divisibility by powers of pp rather than by position on a number line. Two integers are considered close in the p-adic sense when their difference is divisible by a high power of pp.

For a fixed prime pp, the field of p-adic numbers Qp\mathbb{Q}_p is the completion of the rational numbers Q\mathbb{Q} with respect to the p-adic absolute value ∣x∣p=p−vp(x)|x|_p = p^{-v_p(x)}, where vp(x)v_p(x) is the p-adic valuation — the largest power of pp dividing xx. This construction is analogous to how the real numbers R\mathbb{R} arise as the completion of Q\mathbb{Q} under the usual absolute value.

Key Formula

∣x∣p=p−vp(x)|x|_p = p^{-v_p(x)}
Where:
  • pp = A fixed prime number
  • vp(x)v_p(x) = The p-adic valuation of x: the exponent of the highest power of p dividing x
  • ∣x∣p|x|_p = The p-adic absolute value of x

How It Works

The key idea is the p-adic valuation vpv_p. For a nonzero integer nn, vp(n)v_p(n) counts how many times the prime pp divides nn. For a fraction a/ba/b, you compute vp(a/b)=vp(a)−vp(b)v_p(a/b) = v_p(a) - v_p(b). The p-adic absolute value then converts this into a size: ∣x∣p=p−vp(x)|x|_p = p^{-v_p(x)}, so numbers divisible by high powers of pp are small. A p-adic number is represented as a (potentially infinite) expansion in powers of pp going to the right — the opposite direction from decimal expansions.

Worked Example

Problem: Compute the 3-adic absolute value of 45.
Step 1: Factor 45 and find the power of 3.
45=32×5,so v3(45)=245 = 3^2 \times 5, \quad \text{so } v_3(45) = 2
Step 2: Apply the formula for the 3-adic absolute value.
∣45∣3=3−v3(45)=3−2=19|45|_3 = 3^{-v_3(45)} = 3^{-2} = \frac{1}{9}
Answer: ∣45∣3=19|45|_3 = \tfrac{1}{9}. In the 3-adic world, 45 is "small" because it is highly divisible by 3.

Why It Matters

p-adic numbers are central to modern number theory. The Hasse–Minkowski theorem, for instance, solves quadratic equations over the rationals by checking solutions in every Qp\mathbb{Q}_p and in R\mathbb{R}. They also appear in algebraic geometry and cryptography research.

Common Mistakes

Mistake: Assuming larger divisibility by p means a larger p-adic absolute value.
Correction: It is the opposite: higher divisibility by p gives a smaller p-adic absolute value. For example, ∣9∣3=1/9|9|_3 = 1/9 is much smaller than ∣2∣3=1|2|_3 = 1.

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