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For what $(n,k)$ there exists a polynomial $p(x) \in F_2\[x\]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?
Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$

Let $p$ be a prime number. Let $F = \mathbb{Z}/p\mathbb{Z}$. Let $l$ be an odd prime number such that $l \neq p$. Let $X^l - 1 \in F[X]$. Let $K$ be the splitting field of $X^l - 1$. Can we determine the degree $K/F$?

This is a related question.

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marked as duplicate by Brandon Carter, William, Jyrki Lahtonen, Thomas, J. M. Oct 2 '12 at 14:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
Another related question. The question has been discussed in several other questions. Look up cyclotomic cosets for one (but not the only) approach. –  Jyrki Lahtonen Sep 10 '12 at 20:55

1 Answer 1

Let $f$ be the smallest positive integer such that $p^f \equiv 1$ (mod $l$). Let $\Omega$ be the algebraic closure of $F$. Let $\omega \neq 1$ be a root of $X^l - 1$ in $\Omega$. Then, by my answer to this question, the minimal polynomial of $\omega$ over $F$ has degree $f$. Since $F(\omega)$ is the splitting field of $X^l - 1$, The degree of $K/F$ is $f$.

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