Number of prime ideals in quotient ring obtained by $\mathbb {Q}[x]/\langle x^m-1 \rangle$ is ...?
2026-04-04 12:10:34.1775304634
Number of distinct prime ideals in $\mathbb {Q}[x]/\langle x^m-1 \rangle$
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The ideals of $\mathbb Q[x]/(x^m-1)$ correspond to the ideals of $\mathbb Q[x]$ that contain $(x^m-1)$ and these are the ideals generated by the divisors of $x^m-1$.
So the question is reduced to finding the prime decomposition of $x^m-1$. The answer is $$x^m - 1 = \prod_{d\mid m} \Phi_d(x)$$ where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial. It is true but not trivial that $\Phi_n(x)$ is irreducible.
Therefore, the number of prime ideals of $\mathbb Q[x]/(x^m-1)$ is the number of divisors of $m$.