I am quite a beginner with quotient rings. Can someone give me an example of what this quotient ring would look like? $$ \mathbb Z[X]/\phi_m(X) $$
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$\begingroup$ If $f(X)\in \mathbb{Z}[X]$ is a monic irreducible polynomial and $f(\alpha) = 0$ then $\displaystyle\mathbb{Z}[X]/(f(X)) \cong \mathbb{Z}[\alpha] = \{ \sum_{n=0}^{deg(f)-1} c_n \alpha^n, c_n \in \mathbb{Z}\}$ $\endgroup$– reunsCommented Jul 12, 2017 at 1:22
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A concrete model for this ring is $\mathbb Z[\zeta]$, where $\zeta = e^{\frac{2\pi i}{m}}$. This is a subring of $\mathbb C$.
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$\begingroup$ Your first sentence is ambiguous, since $u=1$ satisfies that condition. I’m sure you want to specify that $u$ is to be a primitive $m$-th root of unity. $\endgroup$– LubinCommented Jul 11, 2017 at 14:03
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$\begingroup$ $u$ is a formal symbol. To be more precise we are talking about the polynomial ring $\mathbb Z [x]$ mod the ideal $(x^m -1)$. $\endgroup$– DaronCommented Jul 11, 2017 at 14:23
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$\begingroup$ @Daron, the ring you mention has zero divisors while the one in the question has not. $\endgroup$– lhfCommented Jul 11, 2017 at 14:28