Would someone mind proving this: For all odd primes $p$, $-1\in QR_p \iff p \equiv 1 \pmod 4$.

Question

The proof my textbook gives doesn't make much sense to me.

It says,

$-1 \in QR_p \iff -1^{(p-1)/2} \equiv 1 \pmod p$, then says

If $p \equiv 3 \pmod 4$, then $(-1)^{(p-1)/2} \equiv -1 \neq 1 \pmod p$.

But if $p\equiv 1 \pmod 4$, then $(-1)^{(p-1)/2} \equiv 1 \pmod p$.

I'm not sure about those last two lines, why "If $p \equiv 3 \pmod 4$, then $(-1)^{(p-1)/2} \equiv -1 \neq 1 \pmod p$"?

Thanks!

Answer 1

If $p \equiv 3$ write $p=4k+3,$ then $p-1=4k+2, \ (p-1)/2=2k+1$ is odd, so raising $(-1)$ to that power gives $-1.$ The other case is similar.