Let $\mathbb{F}^n_q$ be a field of $q$ elements, and $poly_D(\mathbb{F}^n)$ be the space of all polynomials in the ring $\mathbb{F}[x_1,...,x_n]$ with degree $\leq{D}$. A Nikodym set is defined as a set $N\subset\mathbb{F}^n$ where for any point $x\in\mathbb{F}_q$, there exists a line $L(x)$ such that $L(x)\backslash{\lbrace{x}\rbrace}\subset{N}$.
We are trying to prove the following: Any Nikodym set in $N\subset\mathbb{F}^n_q$ contains at least $c_nq^n$ elements, where we take $c_n=(10n)^{-n}q^n$.
The proof begins by stating that there exists some non-zero polynomial $P\in{poly_D(\mathbb{F}^n)}$ such that $P$ vanishes on $N$ (the existence of such a polynomial was proven earlier in the text).
This is the portion I didn't understand: Let $x$ be any point of $\mathbb{F}^n_q$. By the definition of a Nikodym set, there is a line $L(x)$ such that $L(x)\backslash{\lbrace{x}\rbrace}\subset{N}$. The polynomial $P$ vanishes on $N$, so $P$ vanishes at $\geq{q-1}$ points of $L(x)$.
I'm not sure how the last line is true, since $L(x)$ is not necessarily the whole field, so how can we say that $P$ vanishing on $N$ implies $q-1$ points?