Cardinality of the ring $ \mathbb{Z}[x]/I $ , $I$ is the ideal generated by $ (x^2+x+1)(x^3+x+1) $

Question

Let $R$ be the ring $ \mathbb{Z}[x]/I $ , Where $I$ is the ideal generated by $ (x^2+x+1)(x^3+x+1) $ and $J$ be the ideal generated by $2$ in $R$.

What is the cardinality of $R$ ?

Answer 1

Updated answer:

The natural embedding $\Bbb{Z}\ \longrightarrow\ \Bbb{Z}[x]$ of $\Bbb{Z}$ as a subring of $\Bbb{Z}[x]$ gives rise to the composition $$\Bbb{Z}\ \longrightarrow\ \Bbb{Z}[x]\ \longrightarrow\ \Bbb{Z}[x]/I.$$ Show that it is injective by proving that $I\cap\Bbb{Z}=\{0\}$.

---

Original answer: (Assuming you meant the cardinality of $R/J$)

HINT: There are canonical isomorphisms \begin{eqnarray*} R/J&=&(\Bbb{Z}[x]/I)/(2\Bbb{Z}[x]/I))\\ &\cong&\Bbb{Z}[x]/(2,(x^2+x+1)(x^3+x+1))\\ &\cong&\Bbb{F}_2[x]/((x^2+x+1)(x^3+x+1)). \end{eqnarray*}