How do finite fields work? I know that a field must satisfy a set of axioms, the one that is causing me most discomfort is closure under addition. All the roots of $X^q-X$ are all the elements of a finite field of order $q$. However, I tested this with $X^5-X$, its roots are ${1,-1,0,i,-i}$ however, if I add $i+i=2i$ and $2i$ is not in the field, failing the axiom of closure, how do finite fields work then? 

Hint: It depends on the definition of the operations. In $\mathbb{Z}_5$, $2+5=7$ but $7=2$ in $\mathbb{Z}_5$.
Edit: Of course 5 as itself is a representative of the equivalence class $[0]$. $0$ is another such representative. I can't understand why people are going pseudo-pedantic about it.
