Question

Show that (2,X) is maximal in Z+XQ[X].

Answer 1

Answer to Question #83127

Let us consider the map $f: Z + xQ[x] \to Z_2$ which maps any polynomial into the residue modulo 2 of the last coefficient. This map is homomorphism as a composition of standard last coefficient homomorphism $Z + xQ[x] \to Z$ and $Z \to Z_2$. $\operatorname{Ker}(f) = <2, x>$. $Z_2$ is a field. By the homomorphism theorem $(Z + xQ[x]) / <2, x>$ is isomorphic to $Z_2$, hence it is a field. So $<2, x>$ is by definition maximal.

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