Question on modular arithmetic and primes

Question

Given a prime $p$ of the form $p = 4k + 1$, with $k$ being an integer, I am trying to prove that $n^{({p-1})/{2}}$ leaves $1$ or $-1$ as remainder when divided by $p$, for any $n \in \{2, \ldots, p-1\}$. Approximately, how many such $n$ are possible for which it leaves $-1$ as remainder?

Answer 1

Hint:

Lil' Fermat asserts that for all such $n$s, $n^{p-1}\equiv 1\mod p$ – inther words, they're all roots of the polynomial $$x^{p-1}=\Bigl(x^{\tfrac{p-1}2}-1\Bigr)=\Bigl(x^{\tfrac{p-1}2}+1\Bigr)$$ Now you can use that a polynomial of degree $d$ over a field cannot have more than $d$ roots in this field to conclude.