Question
Show that if $p$ is an odd prime, then $-1$ is a quadratic residue of $p$ if $p=1(\bmod 4)$, and $-1$ is not a quadratic residue of $p$ if $p$ = 3 (mod 4). [Hint: Use Exercise 62.]
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In this case, we have that $\frac{p-1}{2}$ is even. Show more…
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Show that if $p$ is an odd prime, then $-1$ is a quadratic residue of $p$ if $p \equiv 1(\bmod 4),$ and $-1$ is not a quadratic residue of $p$ if $p \equiv 3(\bmod 4) .[\text {Hint} : \text { Use Exercise } 62 .]$
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Show that if $p$ is an odd prime, then there are exactly $(p-1) / 2$ quadratic residues of $p$ among the integers $1,2, \ldots, p-1$ If $p$ is an odd prime and $a$ is an integer not divisible by $p$ , the Legendre symbol $\left(\frac{a}{p}\right)$ is defined to be 1 if $a$ is a quadratic residue of $p$ and $-1$ otherwise.
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