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p-adic Numbers Nick Raugh March 26, 2008 The p-adic numbers are a device to apply analytic techniques to number theory. Definition 1. An absolute value on a field F is a function with • |−|:F →R • |a| ≥ 0 • |ab| = |a||b| • |a + b| ≤ |a| + |b| In 1918 Ostrowski proved that the only possible absolute values on Q are the trivial, the euclidean, and the p-adic. Definition 2. For a prime p if n ∈ N write n = pα m. With (p, m) = 1. Then |a| |n|p = p1α . Extend this to Q by setting | ab |p = |b|pp . This is an absolute value known as the p-adic absolute value. Any absolute value determines a metric by the formula m(x, y) = |x − y|. The topology induced by this metric is very strange. If |x, y| < p−α , then x ≡ y mod pα . Just as the completion of Q under the usual Euclidean metric is R, we can complete Q with the p-adic metric to get Qp or the the p-adic numbers. Because we have Pn the strong triangle inequality for the p-adic norm wePknow ∞ that any series P i ai converges if and only if ai → 0. For example n n! ∞ n does not. Note that if a ∈ { 0 . . . p − 1 } we know that converges, but n n P∞ P∞ n p−1 n a p converges. For example −1 = p . = (p − 1) n n n 1−p Lemma (Hensel’s Lemma). Let f (x) ∈ Z[x] with f (a) ≡ 0 mod pk and f 0 (a) 6≡ 0 mod p. Then there is a unique b ∈ { 0, . . . , p − 1 } such that f (a + bpk ) ≡ 0 mod pk+1 . it isPenough to define f on Z. Let an = P∞If f :k Znp → Qp is continuous ∞ ∗ (−1) f (n − k). Define f (x) = n an nx . If x an integer then f ∗ = f . k k 1