p-adic Number Arithmetic

p-adic norm · Ultrametric · Ostrowski's theorem · |·|_p vs |·|_∞

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Ultrametric Check

p-adic norm: |n|_p = p^(-v_p(n)) where v_p(n) is the largest power of p dividing n. Numbers divisible by high powers of p are small p-adically. Ostrowski: every non-trivial absolute value on ℚ is either |·|_∞ or |·|_p for some prime p. The ultrametric |x+y|_p ≤ max(|x|_p, |y|_p) makes p-adic geometry strange: every triangle is isoceles.