Generators for polynomials annd multiplicative inverses, not sure what is going on here $X^{-3}$ becomes $X^{12}$. How?

Question

I am confused on this problem. I am not sure what is happening with the (-3) for it to become (12), to then become a larger polynomial in a GF type of problem.

Here is more information about the problem. I hope it helps, but I don't have an actual problem per se. It's more of a concept I want to understand.

Can someone explain why they are doing that calculation with the multiplicative inverses please?

Answer 1

We are in $GF(2^n)$, with $n=4$. It is the splitting field of $x^{16}-x$. Thus $g^{15}=1$ (or $g^{-3}=g^{12}$).