State the reciprocity law for the Jacobi symbol.

Let $a$ be an odd integer $>1$, which is not a square. Prove that there exists a positive integer $n$ such that $n \equiv 1 \bmod 4$ and

$$\left(\frac{n}{a}\right)=-1$$

Prove further that there exist infinitely many prime numbers $p$ such that

$$\left(\frac{a}{p}\right)=-1$$