Number Theory - Elliptic Curves: Number Theory and Cryptography - page 472

then there exists a p-adic integer x with x ≡ x 1 (mod p ) and

f ( x )=0 .

COROLLARY A.3
Let p be an odd prime and suppose b is a p-adic integer that is a nonzero
square mod p.Thenb isthesquareofap-adic integer.

The corollary can be proved by exactly the same method that was used to
prove that − 1 is a square in the 5-adic integers. The corollary can also be
deduced from the theorem as follows. Define f ( X )= X 2 − b and let x 1 ≡ b

(mod p ). Then f ( x 1 ) ≡ 0(mod p )and

f ( x 1 )=2 x 1 ≡ 0(mod p )

since p is odd and x 1 0 by assumption. Hensel's Lemma shows that there

is a p -adic integer x with f ( x ) = 0. This means that x 2 = b , as desired.
When p = 2, the corollary is not true. For example, 5 is a square mod 2 but
is not a square mod 8, hence is not a 2-adic square. However, the inductive
procedure used above yields the following:

≡

PROPOSITION A.4
If b is a 2-adic integer such that b ≡ 1(mod8) then b isthesquareofa 2 -adic
integer.