Polynomial over finite field

Question

I'm currently reading these notes on the simplicity of $PSL_n(F)$. At page 5 it is used that there exists an element x in fields with 4 or more elements such that both:

• $x\neq 0$
• $x^2-1\neq 0$

I have learned, though, that arguments like `a polynomial of degree $n$ has at most $n$ zero points' doesn't work in finite fields. So, why is the above statement true?

Answer 1

If $\forall x\in F:$ $x=0$ or $(x+1)(x-1)=0$, then $x=0$ or $x=1$ or $x=-1$ are the only possible values of $x$ (maybe $1=-1$). But the field has at least four elements...