Chevalley's Theorem reads as follows:
Chevalley's Theorem: if $F(x_1,\cdots,x_n)$ is a form of degree less than $n$, then the congruence $F(x_1,\cdots,x_n)\equiv 0\bmod p$ has a nonzero solution.
Proof. This is a corollary of Proposition 2 here . Because a form of degree $>0$ always has one solution, namely the zero solution, it must have more than one solution, since $n$ is greater than its degree. $\blacksquare$