# Proof for ring isomorphism

Let $p$ be a prime, $\zeta$ a primitive $p$-th root of unity and $\Phi\in\mathbb{Z}[X]$ the $p$-th cyclotomic polynomial (i.e. $\Phi = 1 + X + X^2 + \ldots + X^{p-1}$). Let further be $k\in\mathbb{Z}$. Look at the mapping $$\varphi : \mathbb{Z}/\Phi(k) \to \mathbb{Z}[\zeta]/(k-\zeta),\quad z + (\Phi(k)) \mapsto z+(k-\zeta)$$ By direct and boring computations, it can be shown that $\varphi$ is a well-defined isomorphism of rings.

Somehow I have the strong feeling that these computations are not strictly necessary. There should be a more abstract top-down approach, using for example isomorphism theorems, the property of the evaluation homomorphism etc. However, I was not able to figure out how to do it. Any ideas?

They're both isomorphic to $\mathbb{Z}[X] / \langle \Phi(X), X-k \rangle$:
$$\mathbb{Z}[X] / \langle \Phi(X), X-k \rangle \cong (\mathbb{Z}[X] / \Phi(X))/(X-k) \cong \mathbb{Z}[\zeta] / (\zeta - k)$$
$$\mathbb{Z}[X] / \langle \Phi(X), X-k \rangle \cong (\mathbb{Z}[X] / (X-k))/\Phi(X) \cong \mathbb{Z} / \Phi(k)$$