Determine the degree [K:F] of the extension K over F, where K is a splitting field for the given polynomial:
i)
ii)
I can do these type of problems for the case F=Q, but how do you go about it with a finite field such as Z_2?
The splitting field of over is where is a primitive -th root of unity (we know such a primitive root exists because where ). The polynomial can be factored into irreducibles as . Therefore, has degree over which means .
There happens to be a more general theorem about cyclotomic extensions of finite fields but I do not want to say it because I think it will just lead to your confusion, I think it would be easier for you to try to understand the solution in the above paragraph.