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Math Help - Splitting field over Z_2

  1. #1
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    Splitting 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) F = Z_2, x^7 -1
    ii) F = Z_2, x^7 -1

    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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  2. #2
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    Quote Originally Posted by Amanda1990 View Post
    Determine the degree [K:F] of the extension K over F, where K is a splitting field for the given polynomial:

    i) F = Z_2, x^7 -1
    ii) F = Z_2, x^7 -1

    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 x^7 - 1 over F is K = F(\zeta) where \zeta is a primitive 7-th root of unity (we know such a primitive root exists because 2\not | 7 where 2=\text{char}(F)). The polynomial x^7 - 1 can be factored into irreducibles as (x-1)(x^3+x+1)(x^3 + x^2+1). Therefore, \zeta has degree 3 over F which means [K:F]=3.

    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.
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