- the group of characters was a cyclic group of order $p-1$.
- Legendre symbol was a character of order $2$.
Thus by the uniqueness of the order $2$ elements in cyclic group, all second order characters were Legendre.
However, I was wondering if there was any other way to prove it. For example, prove it without the knowledge of the first statement.
Let $p$ be an odd prime. Consider the multiplicative group $G=\Bbb F_p^\times$. The map $s:G\to G$ defined by $s(a)=a^2$ is a homomorphism with kernel $\{\pm1\}$. Its image $s(G)$ therefore has order $|G|/2=(p-1)/2$. Any character $\chi:G\to\{\pm1\}$ must be trivial on $s(G)$ as that consists of squares. Thus $\chi$ factors through $G\mapsto G/s(G)$ and as $|G/s(G)|$ has order $2$, there is only one character of order $2$ on $G$. This must be the Legendre symbol.