1. ## Determine solvablity

Prove that $x^{2}+2\equiv0 \mod p$ is unsolvable if $p\equiv 5,7\mod 8$ .
Thanks

2. Originally Posted by chipai
Prove that $x^{2}+2\equiv0 \mod p$ is unsolvable if $p\equiv 5,7\mod 8$ .
Thanks
Can we use Gauss's Lemma ?

To determine whether the given number $a \neq 0$ is quadratic residue , consider the list :

$a , 2a , 3a , ..., Pa ~,~ P = \frac{p-1}{2}$

If the number of the elements in the above list lying between $P+1$ and $2P$ inclusively is even , then $a$ is quadratic residue , otherwise , it is quadratic non-residue .

In this case , $a \equiv -2 \bmod{p}$ and $P$ is either $2\bmod{4}$ or $3 \bmod{4}$

The elements lying between the range should be :

$-2 , -4 , .... , (-2) \frac{P}{2}$ for $P \equiv 2 \bmod{4}$ so the number is $\frac{P}{2}$ an odd number .

Or

$-2 , -4 , ..., (-2) \frac{P-1}{2}$ for $P \equiv 3 \bmod{4}$ so the number is $\frac{P-1}{2}$ also an odd .

Therefore , $- 2$ is not the quadratic residue of modulo $p \equiv 5,7 \bmod{8}$ .

Remarks : so we can see that $- 2$ is quadratic residue of modulo $p \equiv 1,3 \bmod{8}$ .