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? 

The main point: the elements of a finite field $Z_p$ ($p$ - prime) are $0, 1, ..., p-1$ (or more precisely, equivalence classes modulo $p$...). Numbers from $\mathbb{C}$ or $\mathbb{R}$ are not in $Z_p$.
For every element of $Z_5$, $x^5-x \equiv 0 \bmod 5$ is true. You can check it yourself. Remember that all field operations in $Z_p$ are modulo $p$, $p=5$ in you case. For instance $2^5 \equiv 2 \bmod 5$.
"Finite Fields" by Rudolf Lidl, Harald Niederreiter is an excellent book on this topic.
