I will leave a comment if its not correct Let p be a prime s

I will leave a comment if its not correct.

Let p be a prime such that p Congruent 3 (mod 4). Show that equation x^2 Congruent -1 (mod p) has no solutions.

Solution

Proof by contradiction:

Suppose x² -1 (mod p).

Raise both sides to the (p - 1)/2 power to obtain

x^p-1 (-1)^(p-1)/2 (mod p).

By Fermat, x^p-1 1. Since p 3 (mod4),

the exponent (p - 1)/2 is odd.

Therefore

(-1)^(p-1)/2 = -1. This yields 1 -1 (mod p),

which is a contradiction.

Therefore x cannot exist.