Suppose that is a zero of in some field extension of . Write as a product of linear factors in .
Since is a zero, is a factor of . My plan was to simply divide into to obtain a cubic and then find a root of the cubic, etc. until I obtained all the linear factors. Is there a reason that shouldn't divide ? I don't see one but it is not dividing it when I try. Does the fact that lies in and not affect this?
I got it. Since we're in characteristic 2 we have for arbitrary (I did work out the details).
Question about something I'm not clear on though... My book (Contemporary Abstract Algebra, Gallian, 5th ed.) says "A field is an extension field of a field if and the operations of are those of restricted to ." That wording bothers me... Is that saying that E inherits the operations of F (and thus has the same characteristic?)? That's not what it seems to be saying to me, but it's what seems to make sense.