# The order of the Galois group of a cyclotomic field over a finite prime field [duplicate]

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 was marked as an exact duplicate of an existing 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

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 $f$.