g = X^3 + X + 1

Let E be the extension of F2 with a root alpha of g. Write down a complete list of elements of E without repetitions.

I've tried by best at this question, but I still don't feel that I fully understand what I'm doing, (which will probably be proved with it being wrong...)

We know is a route of g, so it follows that is also a root of g. We can apply this repeatedly to find roots and so on.

Hence

Hence

Hence the roots are

Anyone know if this is OK?

Thanks in advance