What is the isomorphism between $\mathbb Z[x]/(2,x)$ and $\mathbb Z/2 \mathbb Z$?

Question

I am trying to show that $\mathbb Z[x]/(2,x)$ is a field by showing that it is isomorphic to $\mathbb Z/2 \mathbb Z$ but I do not know how to write this isomorphism explicitly. could someone help me in this please?

Answer 1

Just unwind the following isomorphisms: $$\newcommand{\Z}{\Bbb Z} \Z[x]/(2,x) \color{red}{\cong} \frac{\Z[x]/(x)}{(2,x)/(x)} = \frac{\Z[x]/(x)}{(2+(x))} \color{blue}{\cong} \frac{\Z}{(2)} = \Z/2\Z, $$ where ‘$\color{red}{\cong}$’ is by the third isomorphism theorem, and ‘$\color{blue}{\cong}$’ follows from the isomorphism $\Z[x]/(x) \to \Z$ which induces the map $\Z[x] \to \Z$ defined by $p \mapsto p(0)$.

Answer 2

The kernel of the surjective morphism $$\mathbb Z[X]\to\mathbb Z/2 \mathbb Z,\quad P(X)\mapsto\overline{P(0)}$$ is $(2,X).$