We build a finite field $\mathbb Z/p\mathbb Z$, $p>2$. Then we introduce a sign function, which is $0$ at $0$, $+1$ at $1\dots(p-1)/2$, and $-1$ at $(p+1)/2\dots p-1$. Now we want to generalize the function onto the algebraic closure of the field.
The straightforward way to go is to represent the sign function is a polynomial of degree $p-2$. In addition to $x=0$, this polynomial will have $p-3$ roots in some field extension.
Now if we consider different $p$, will there be any system here? Like, what the roots are, what field extension they generate, etc.? Is there anything special about these polynomials?
Is it the only way to introduce a sign function? What are other interesting ways?
