Why $1$ is the only quadratic residue modulo $8$? I'm trying to understand the Proposition 5.1.1 - Ireland and Rosen, A Classical Introduction to Modern Number Theory, p.50, however, I can't understand why this argument is true:
$1$ is the only quadratic residue mod $8$.
I wrote a program to generate all quadratic residue modulo $8$, from $0$ to $7$
0 -> 0
1 -> 1
4 -> 4
9 -> 1
16 -> 0
25 -> 1
36 -> 4
Press any key to continue . . .

I saw $4$ there, so how come only $1$ satisfied?  
The original text was,

Thank you,
@Bill Dubuque: Thank you for the reference.

Quadratic residues are usually taken from the unit group. $2$ is not in $U_8$.
The wikipedia article mentions this issue, citing Ireland and Rosen.
