I am a beginner in abstract algebra and I am trying to prove $\mathbb{Z}[X]/(2,X) \simeq \mathbb{Z}/2\mathbb{Z}$.
Let $\varphi : \mathbb{Z}[X] \rightarrow \mathbb{Z}/2\mathbb{Z}$. $\varphi$ is a homomorphism ring. We get the isomorphism if :
- $\ker \varphi = (2,X)$ and
- $\mathrm{Im} \varphi = \mathbb{Z}/2\mathbb{Z}$.
But I had a few problems. I don't know if what I did is correct.
Proof of 1.
Let $P(X) : = \sum_{i = 0}^{+ \infty} a_iX^i \in \ker \varphi$ with $a_i \in \mathbb{Z}$. So $\varphi(P(X)) = \overline{0}$ because $\overline{0}$ is the zero-element of $\mathbb{Z} / 2 \mathbb{Z}$.
Since $\varphi$ is a homomorphism ring, then we get $\varphi(a_i X^i) = \overline{a_i X^i} = \overline{0}$. Here we get $\overline{a_i} = \overline{0}$ or $\overline{X^i} = \overline{0}$ i.e $ \ker \phi$ is generated by $0$ and $X$.
I get $0$ instead of $2$, I don't know why.
Proof of 2.
Let $\overline{k} \in \mathrm{Im} \varphi$, with $k = 0,1$, then $\exists P(X) \in \mathbb{Z}[X]$ s.t $\varphi(P(X)) = \overline{k}$
And I don't know how to pursue. Thank you.
Hint
The ideal
$$I=(2,x)=\{2n +xq(x) \mid n \in \mathbb Z, \, q \in \mathbb Z[x]\}$$
is the kernel of the ring homeomorphism that maps $p \in \mathbb Z[x]$ to $\overline{p_0} \in \mathbb Z/2 \mathbb Z$ where $p_0$ is the coefficient of degree $0$ of $p$.