Number Theory - Elliptic Curves: Number Theory and Cryptography - page 470

The p -adic integers are most easily regarded as sums of the form

∞

a n p n , n ∈{ 0 , 1 , 2 ,...,p− 1 }.

n =0

Such infinite sums do not converge in the real numbers, but they do make
sense with the p -adic absolute value since a n p n .

Arithmetic operations are carried out just as with finite sums. For example,
in the 3-adic integers,

| | p → 0as n →∞

(1 + 2 · 3+0 · 3 2 + ··· )+(1+2 · 3+1 · 3 2 + ··· )=2+4 · 3+1 · 3 2 + ···
=2+1 · 3+2 · 3 2 + ···

(where we wrote 4 = 1 + 3 and regrouped, or “carried,” to obtain the last
expression). If

x = a k p k + a k +1 p k +1 + ···

with a k =0,then

−x =( p − a k ) p k +( p − 1 − a k +1 ) p k +1 +( p − 1 − a k +2 ) p k +2 + ··· (A.1)

(use the fact that p k +1 +( p 1) p k +1 +( p 1) p k +2 + = 0 because the sum

telescopes, so all the terms cancel). Therefore, p -adic integers have additive
inverses. It is not hard to show that the p -adic integers form a ring.
Any rational number with denominator not divisible by p is a p -adic integer.
For example, in the 3-adics,

− − ···

1
2 =

− 1

1 − 3 = − (1+3+3 2 + ··· )=2+3+3 2 + ··· ,

where we used (A.1) for the last equality. In fact, it can be shown that if

x = n =0 a n p n is a p -adic integer with a 0 =0,then1 /x is a p -adic integer.
The p -adic rationals , which we denote by Q p , are sums of the form

y = ∞ a n p n , (A.2)

n = m

with m positive or negative or zero and with a n ∈{ ∈ Q p ,

then p k y is a p -adic integer for some integer k .The p -adic rationals form a
field, and every rational number lies in Q p .If a m

0 , 1 ,...,p − 1 } .If y

= 0 in (A.2), then we define

|y| p = p −m .

v p ( y )= m,

This agrees with the definitions of the p -adic valuation and absolute value
defined above when y is a rational number.
Another way to look at p -adic integers is the following. Consider sequences
of integers x 1 ,x 2 ,... such that

(mod p m ) x m ≡ x m +1 (A.3)