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?

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- April 14th 2009, 08:06 AMAmanda1990Splitting field over Z_2
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? - April 14th 2009, 11:00 AMThePerfectHacker
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.