I have looked all over but I can't really find a clear example of this...
If $\mathbb{Z}_5$ is set $\{0,1,2,3,4\}$ prove that it is a field... I understand that from the table we can see that the set is commutative, associative, and it has an identity. I'm not sure how to show the inverse $-a$ is in $F$, but by the table we can see that for take $2$ is in $F$ then if the inverse exists $-2$ is in $F$, and $2 + (-2) = 0$ but in the table we can see that $2 + 3 = 0$ so then $-2 = 3$? But I am just confused how that is possible, I know maybe it has something to do with mod but not sure how to tie it all together table
$\mathbb{Z}_5$ is a set "equivalence classes": we consider two integers $a$ and $b$ (such as $-2$ and $3$) to be "equivalent modulo $5$" if, divided by $5$, they have the same remainder. So $$-2=5\cdot (-1)+\underbrace{3}_{\text{remainder}}\quad\text{and}\quad 3=5\cdot 0+\underbrace{3}_{\text{remainder}}\implies -2\equiv 3 \pmod{5}.$$ In a similar way, looking at your table we read that $3\cdot 4$ yields $2$: $$3\cdot 4=12=5\cdot 2+\underbrace{2}_{\text{remainder}}\equiv 2\pmod{5}.$$