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