1. ## Law of Quadratic reciprocity2

Let p be an odd prime number. Prove that

(3 over p) =1 if and only if p = +-1 mod 12

where (3 over p) denotes the legendre symbol.

2. Originally Posted by mndi1105
Let p be an odd prime number. Prove that

(3 over p) =1 if and only if p = +-1 mod 12

where (3 over p) denotes the legendre symbol.
Let $p>3$. There are two cases: $p\equiv 1(\bmod 4)$ or $p\equiv 3(\bmod 4)$. In the first case we get $(3/p) = (p/3)$. Now if $p\equiv 1(\bmod 3)$ then $(p/3) = (1/3)=1$ and if $p\equiv 2(\bmod 3)$ then $(p/3) = (2/3) = -1$, therefore $p\equiv 1(\bmod 3)$ in the first case. Together $p\equiv 1(\bmod 4)$ and $p\equiv 1(\bmod 3)$ give us $p\equiv 1(\bmod 12)$.

In the second case we get $(3/p) = -(p/3)$ and to get $(3/p) = 1$ it is necessary and sufficient to get $(p/3) = -1$. Now this happens when $p\equiv 2(\bmod 3)$. We have $p\equiv 3(\bmod 4)$ and $p\equiv 2(\bmod 3)$ which is equivalent to $p\equiv -1(\bmod 4)$ and $p\equiv -1(\bmod 3)$. Together this combines into $p\equiv -1(\bmod 12)$.