Show that the ideal $(2,f(x)) \subset \mathbb Z[x]$ is maximal for a given condition.

Question

Let $\phi_2 : \mathbb Z[x] \to \mathbb Z_2[x]$ defined by $\phi_2(\sum_{i=0}^{n}a_ix^i)=\sum_{i=0}^{n}{\bar a_i}x^i$, where $\bar {a_i} :=a_i \ mod \ 2.$

Show that the ideal of the form $(2,f(x)) \subset \mathbb Z[x]$ such that $\phi_2(f(x))$ is irreducible are maximal ideals.

How can I show it?

Basically if we get a onto homomorphism from $\mathbb Z[x] $ to a field whose kernel is $(2,f(x))$ then it must be maximal ideal. Otherwise we have to show applying definition.

Please someone give some hints.

Thank you..

Answer 1

Hint: $$\mathbb{Z}[x] \xrightarrow{\phi_2} \mathbb{Z}_2[x]\xrightarrow{\pi}\mathbb{Z}_2[x]/(\phi_2(f(x))).$$ Here $\mathbb{Z}_2[x]\xrightarrow{\pi}\mathbb{Z}_2[x]/(\phi_2(f(x)))$ is the natural quotient map.

Answer 2

Ideal $I$ is maximal in commutative ring $R$ if and only if $R/I$ is a field.

We have $\mathbb Z[x]/(2,f(x))\cong \mathbb Z_2[x]/(\phi_2(f(x))).$ Can you finish?