$\mathbb Z [X] / (X)$ isomorphic to $\mathbb Z[X] / (X+1)$ isomorphic to $\mathbb Z [X] / (X+2015)$

Question

I have to prove that $\mathbb Z [X] / (X) \cong \mathbb Z[X] / (X+1) \cong \mathbb Z [X] / (X+2015)$.

I think that one answer could be that $\mathbb Z[X]/(X) \cong \mathbb Z(0)$, $\mathbb Z[X]/(X+1)\cong Z(-1)$ and $\mathbb Z [X] /(X+2015)\cong \mathbb Z(-2015)$, as $0, -1, -2015$ are roots of these polynomials. Also, as $0, -1, -2015\in \mathbb Z$, then all of them are isomorphic to $\mathbb Z $, hence isomorphic between them.

But I don't know very well how to justify this rigorously and my teacher has said to me that there is a simpler way to proof this, without using the roots of these polynomials.

Answer 1

Some ideas:

Define

$$\phi:\Bbb Z[x]\to\Bbb Z[x]/\langle x+1\rangle\;,\;\;\phi(p(x)):=p(x+1)+\langle x+1\rangle$$

Show that this is a homomorphism, and now: what is its kernel? You may want to apply the first isomorphism theorem.