Why does the Legendre symbol $(\frac {p}{3}) = 1 $ if $p \equiv 1 \pmod 3$ and $-1$ if $p \equiv 2 \pmod 3$?

Question

Why does the Legendre symbol

$(\frac {p}{3}) = 1$ if $p \equiv 1 \pmod 3$ and $-1$ if $p \equiv 2 \pmod 3$?

I understand how to show this for $(\frac {2}{p})$ and $(\frac {-1}{p})$ but for some reason this one I don't understand.

Answer 1

Recall that if $q$ is prime, and $a$ is not divisible by $q$, then $a$ is a QR of $q$ if there exists an $x$ such that $x^2\equiv a\pmod{q}$.

Now if $b\equiv a\pmod{q}$, it follows that $b$ is a QR of $q$ if and only if $a$ is a QR of $q$. For the $x$ such that $x^2\equiv a\pmod{q}$ equally well satisfies $x^2\equiv b\pmod{q}$.

Now we are finished, for it is clear that $1$ is a QR of $3$ and $2$ is an NR of $3$. So if $b\equiv 1\pmod{3}$, then $b$ is a QR of $3$, and if $b\equiv 2\pmod{3}$ then $b$ is an NR of $3$.

Remark: Please note that this is a low level result, a direct consequence of the definition of QR and NR. It has no direct connection to Reciprocity, though it is used routinely in Legendre symbol calculations.