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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:

  1. 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.

  2. 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 $0 < k < n$.

  3. 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 $\{\zeta, \zeta^2, \ldots , \zeta^n\} = U_n$.

  4. 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 $\mathbb{Q}(\zeta) = \mathbb{Q}(\zeta_1, \ldots , \zeta_n)$.

  5. 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 $x^n -1 \in \mathbb{Q}_n[x]$?

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1 Answer 1

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Because $\mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ is a splitting field for $x^n - 1$.

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But it's a splitting field for $x^n - 1$ over $\mathbb{Q}$. How do we know we're not getting more elements than necessary as a result of adding the extra elements from $\mathbb{Q}$? That is, how do we know that $\mathbb{Q}(\zeta_1, \ldots , \zeta_n)$ is a splitting field for $x^n - 1 \in R_n[x]$ and not just $x^n - 1 \in \mathbb{Q}[x]$? –  user1770201 Oct 3 '13 at 18:18
1  
@user: Ah, I missed $R_n$. Actually, what you describe is extremely weird, because $R_n$ isn't even a ring! And most of the coefficients of $x^n - 1$ are zero, which aren't even elements of $R_n$. Can you double check that you have the problem correct? –  Hurkyl Oct 3 '13 at 19:43
    
You're right -- the question wasn't written down correctly. –  user1770201 Nov 8 '13 at 16:14

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