Dummy question related to first isomorphism theorem

Question

I would like to show that $\mathbb Z[x]/\left<x\right>$ is isomorphic to $\mathbb Z[x]/\left<x-1\right>$, and hence show that $\mathbb Z[x]/\left<x-1,p\right>$ is isomorphic to $\mathbb Z_p$.

I am completely new to ring theory. Could someone illustrate for me step-by-step

1. How can I define a map?

2. How can I show that the map is surjective?

3. How can I apply the first isomorphism theorem?

Why is $\mathbb{Z}[x]/(1-x,p)$ isomorphic to $\mathbb{Z}_{p}$, where $p$ is a prime integer.

How to show $\mathbb{Z}[x]/\left<2,x\right>$ is isomorphic to $\mathbb{Z}_2$

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

These are proofs related to my questions, but I am not able to understand them so far...

Thank so much!!!

Answer 1

Let's try the following. Define:

$$\begin{cases}\phi:\Bbb Z[x]\to\Bbb Z\;,\;\;\phi(f(x)):=f(0)\\{}\\ \psi:\Bbb Z[x]\to\Bbb Z\;,\;\;\psi(f(x)):=f(1)\end{cases}$$

Check both maps above are ring homomorphisms, they both are surjective, and also

$$\ker\phi=\{f(x)\in\Bbb Z[x]\;/\;f(0)=0\}=\langle x\rangle\;,\;\;\ker\psi=\{f(x)\in\Bbb Z[x]\;/\;f(1)=0\}=\langle x-1\rangle$$

and now apply the first isomorphism theorem (rings version) to the above to get isomorphisms

$$\Bbb Z[x]/\langle x\rangle\cong\Bbb Z\;,\;\;\Bbb Z[x]/\langle x-1\rangle\cong\Bbb Z$$

so now just compose one of the above isomorphisms with the inverse map of the other one to get what you want:

$$\Bbb Z[x]/\langle x\rangle\cong\Bbb Z[x]/\langle x-1\rangle$$

For the rest, you can show first that

$$\Bbb Z[x]/\langle 1-x,\,p\rangle\cong\left(\Bbb Z[x]/\langle \,p\,\rangle\right)/\left(\langle1-x,\,p\rangle/\langle\,p\,\rangle\right)$$

using the third (or second or something) isomorphism theorem, and then show

$$\begin{cases}\Bbb Z[x]/\langle \,p\,\rangle\cong\Bbb Z_p[x]\\{}\\\langle1-x,\,p\rangle/\langle\,p\,\rangle\cong\langle1-x\rangle\end{cases}$$

Further hints: check the maps

$$f(x)\mapsto f(x)\pmod p\;,\;\;f(x)(1-x)+g(x)p\mapsto f(x)$$