Division in finite fields Let's take $GF(2^3)$ as and the irreducible polynomial $p(x) = x^3+x+1$ as an example. This is the multiplication table of the finite field
I can easily do some multiplication such as $$(x^2+x)\cdot(x+1) = x^3 + x^2 + x +1 = x+1+x^2+x+2 = x^2$$
I am wondering how to divide some random fields such as $x^2 / (x+1)$. The result is $x^2+x$ (compare above). But how do I actually calculate this. Polynomial long division does not help be:



*

*Why don't I get $x+1$ as result? 

*How can I calculate $x / (x^2+x+1)$? The result should be $x+1$


Hint
You just go the other way around. If $a = bc$ then b and c are on the row and column and a is inside the table. So if $a/c = b$ the same should hold. Now see what c and a and b are for your example and go check in the table.

So for example 2) you check which row will give you $x$ inside the table for the column with $x^2+x+1$ on top.
