Number Theory proving questions

Question

Let $n$ be a positive integer such that $n \equiv 3 \pmod 4$. Prove that

$x^2 \equiv -1 \pmod n$

is not solvable for integer $x$.

Answer 1

Let $n$ be an integer of the form $4k+3$ then it's divisible by a prime $p$ of the form $4k+3$, otherwise all its divisor will be of the form $4k+1$ and it will be also of this form.

Let $p=4k+3$ such a prime , and assume that there is some integer $x$ such that $x^2\equiv -1 \mod n\equiv -1\mod p$ then we have $x^{p-1}\equiv (x^2)^{\frac{p-1}{2}}\equiv (-1)^{\frac{p-1}{2}}\equiv (-1)^{2k+1}\equiv -1 \mod p$

now using Fermat's little theorem we have $x^{p-1}\equiv 1 \mod p$ which is absurd.