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 http://thestudentroom.co.uk/latexren...2c28b24630.png is a route of g, so it follows that http://thestudentroom.co.uk/latexren...7777bc68a8.png is also a root of g. We can apply this repeatedly to find roots http://thestudentroom.co.uk/latexren...0ed4554cd3.png and so on.

http://thestudentroom.co.uk/latexren...c9c26af62b.png

Hence http://thestudentroom.co.uk/latexren...e623e86387.png

Hence http://thestudentroom.co.uk/latexren...a4218267bb.png

Hence the roots are http://thestudentroom.co.uk/latexren...7ae20cf77b.png

Anyone know if this is OK?

Thanks in advance http://static.thestudentroom.co.uk/i...lies/smile.gif