Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$
$\alpha^4=\alpha+1$
$\alpha^5=\alpha^2+\alpha$
... (the rest are omitted - all the way up to) ...
$\alpha^{14}=\alpha^3+1$
$\alpha^{15}=1$
Now we are asked to find $(\alpha^{38})^{-1}$ in $\mathbb F$
$\alpha^{38} \equiv \alpha^8$ in $\mathbb Z_2[x]$ I think? and $\alpha^8$ happens to be $\alpha^2+1$
and an Inverse of an element $g$ in $\mathbb Z_2[x]$ means to find $g^{-1}$ such that ($g\times g^{-1}) \equiv 1$ in $\mathbb Z_2[x]$ right?
So am I correct in that what we need to find is the inverse of $\alpha^2+1$ such that $(\alpha^2+1)(\alpha^2+1)^{-1} \equiv 1$ in $\mathbb Z_2[x]$? And also, how do you do that?