Let $p,q$ be odd rational primes and $n=pq$, $\zeta_{n}$ a primitive $n$th root of unity, $K = \mathbb{Q}(\zeta_n)$ the $n$th cyclotomic field and $\mathcal{O}_{K} = \mathbb{Z}[\zeta_n]$ its ring of integers.
Are the ideals $I = (\zeta^{p}_{n} -1), J = (\zeta^{q}_{n} -1)$ prime in $\mathcal{O}_{K}$? How would you prove this?
Best Answer
$$\Phi_p(X)= \sum_{m=0}^{p-1} X^m, \qquad \#\Bbb{Z}[\zeta_5]/(1-\zeta_5)=N_{Q(\zeta_5)/Q}(1-\zeta_5) = \prod_{k=1}^4(1-\zeta_5^k) = \Phi_5(1) = 5$$ Thus $1-\zeta_5$ is prime in $\Bbb{Z}[\zeta_5]$.
Then for $p \nmid 5$ because $\Phi_p(X)$ is irreducible over $\Bbb{Q}(\zeta_5)$
$$\Bbb{Z}[\zeta_5,\zeta_p]/(1-\zeta_5)\cong\Bbb{Z}[\zeta_5][X]/(\Phi_p(X))/(1-\zeta_5)\cong\Bbb{Z}[\zeta_5]/(1-\zeta_5)[X]/(\Phi_p(X))$$ $$\cong \Bbb{Z}/(5)[X]/(\Phi_p(X))\cong \Bbb{Z}[X]/(\Phi_p(X))/(5)\cong \Bbb{Z}[\zeta_p]/(5)$$ and hence $(1-\zeta_5)$ is prime in $\Bbb{Z}[\zeta_5,\zeta_p]$ iff $(5)$ is prime in $\Bbb{Z}[\zeta_p]$ iff $5$ is of order $p-1$ modulo $p$.