Consider a field $k$ of characteristic $0$ and a positive integer $n.$ In the proof of Theorem 4.19 of Polytopes, Rings, and K-Theory by Bruns and Gubeladze, it is stated that we have an isomorphism $k[\mathbb Z / n \mathbb Z] \cong k[x] / (x^n - 1),$ where $k[\mathbb Z / n \mathbb Z]$ is the group ring corresponding to the cyclic group of integers modulo $n;$ however, I am having difficulty convincing myself of this. I believe that the $k$-algebra homomorphism $\varphi : k[x] \to k[\mathbb Z / n \mathbb Z]$ induced by the assignment $x^m \mapsto \overline m$ is surjective, where we denote $\overline m = m \text{ (mod } n),$ so I would like to show that $\ker \varphi = (x^n - 1),$ but I have not been able to do this.
I would greatly appreciate any insight, comments, or suggestions. Thank you.
About the first question. The kernel contains $x^n-1$. The polynomial ring is PID so the kernel is generated by some polynomial $f(x)$. Then the remainder $r(x)$ of $f$ when divided by $x^n-1$ must belong to the kernel. That $r(x)$ has degree $<n$. Easy linear algebra then shows that $r$ is $0$. Hence the kernel is generated by $x^n-1$.