TITLE: Show that $\mathbb{Q}(\zeta)$ is the Splitting Field for $x^n - 1 \in R_n[x]$
QUESTION [3 upvotes]: Let $R_n = \{\bar{x}$ modulo $n : (x,n) = 1\}$ which forms a group under multiplication. Let $p(x) = x^n - 1 \in \mathbb{Q}_n[x]$ have roots $\zeta_1, \zeta_2, \ldots , \zeta_n$. 
Prove that there is a $\zeta \in \{\zeta_1, \ldots , \zeta_n\}$ s.t. $\mathbb{Q}(\zeta)$ is the splitting field for $p(x)$.
Attempt:

Each $\zeta_i \in \{\zeta_1, \ldots , \zeta_n\}$ satisfies $(\zeta_i)^n - 1 = 0$ and hence satisfies $(\zeta^i)^n = 1$ and hence is an $n$th root of unity.
The set $U_n = \{\zeta_1, \ldots , \zeta_n\}$ forms the cyclic group of $n$th roots of unity.  That is, each $\zeta_i = e^{k i 2 \pi/n}$ for some $0 < k < n$.  For ease of notation, set each $\zeta_k = e^{k i 2 \pi/n}$ for all $0 < k < n$.
Since $U_n$ is cyclic, let $\zeta \in U_n$ be a generator for $U_n$ and consider that $\mathbb{Q}(\zeta) \supseteq U_n$ since $\zeta, \zeta^2, \ldots , \zeta^n$ are all $\mathbb{Q}$ polynomials with coefficient $1$ and $\{\zeta, \zeta^2, \ldots , \zeta^n\} = U_n$.
Then $\mathbb{Q}(\zeta) \supseteq \mathbb{Q}(\zeta_1, \ldots , \zeta_n)$, and since we know trivially that $\mathbb{Q}(\zeta) \subseteq \mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ we have that $\mathbb{Q}(\zeta) = \mathbb{Q}(\zeta_1, \ldots , \zeta_n)$.
Then since we know $\mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ is the minimal field containing $\zeta_1, \ldots , \zeta_n$ and $\mathbb{Q}$, we know that $\mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ is the splitting field for $x^n - 1 \in \mathbb{Q}[x]$.  Hence $\mathbb{Q}(\zeta)$ is also the splitting field for $x^n - 1 \in \mathbb{Q}[x]$.

Question: How do I go from (4) to obtain that $\mathbb{Q}(\zeta)$ is the splitting field for $x^n -1 \in \mathbb{Q}_n[x]$?

REPLY [1 votes]: Because $\mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ is a splitting field for $x^n - 1$.
{"set_name": "stack_exchange", "score": 3, "question_id": 513704}