About the variable of the Chebyshev polynomials over finite fields

Question

I come across the following terminology: Chebyshev polynomials over finite fields, they are used in graph theory. They have the same recurrence equation and other properties. I am confused about the variable $x$ in their definition. Is it the ordinary real variable in $[-1,1]$ or the $x$ is an element of the finite field.

Answer 1

If $\mathbb F$ is a field, and $p(x)\in \mathbb F[x]$ is any polynomial with coefficients in $\mathbb F$ then you can certainly evaluate $p(x)$ at any element in that field or in any extension of that field.

Thus, if $p(x)=x^2+1$ is defined over $\mathbb Q$, it would make sense to evaluate it on any complex number as $\mathbb Q$ is a subfield of $\mathbb C$. Similarly, if $p(x)$ is defined over a finite field it would make sense to evaluate it on an element of that field or any extension of that field.