# Math Help - Maximal Ideal

1. ## Maximal Ideal

Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].

2. Originally Posted by Coda202
Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].
I don't beleive it is a maximal Ideal... Here is why

Define a Homomorphism from $\phi:\mathbb{Z}[x] \to \mathbb{Z}$ by $\phi(f) \to f(0)$

Note that the kernel of $\phi$ is N

$\phi$ is onto so we can use the first isomorphism theorem to get

$\mathbb{Z}/N \cong \mathbb{Z}$

Since $\mathbb{Z}$ is an Integral Domain this implies that The Ideal is Prime and not maximal.

This ideal in $\mathbb{Z}[x]$ is contained in the Ideal gentered by (2, and the same set f) and this ideal is maximal