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Thread: 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) $\displaystyle F = Z_2, x^7 -1$
    ii) $\displaystyle F = Z_2, x^7 -1$

    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) $\displaystyle F = Z_2, x^7 -1$
    ii) $\displaystyle F = Z_2, x^7 -1$

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

    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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