basic question about field theory
I know that if $p(x)$ is an irreducible polynomial in $\mathbb{Q}$, then $$\frac{\mathbb{Q}[x]}{\langle p(x) \rangle}$$ is a field.
But if $p(x)$ is irreducible over $\mathbb{Z}$, then we have $$\frac{\mathbb{Z}[x]}{\langle p(x) \rangle}$$
Now I am wondering that how these two fields differ?
I am extremely confused because the polynomial $x^3+9x+6$ is irreducible over $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ so we have the corresponding quotient fields.
If $\theta$ is a root of $p(x)$ then the element $1+\theta$ as an inverse $\frac{\theta^2-\theta+10}{4}$ in $$\frac{\mathbb{Q}[x]}{\langle p(x) \rangle}$$
but it doesn't lie in $$\frac{\mathbb{Z}[x]}{\langle p(x) \rangle}$$
But both of them are fields. So that element must have inverse in both of them Probably I am missing something but extremely confused.
Irreducibles in UFDs generate prime ideals (and not necessarly maximal ideals), so the quotient in general are integral domains. For your specefic example of $\mathbb{Z}[X]$, maximal ideals have the form $\langle n,p \rangle$, where $n$ is a prime integer and $p$ is a primitive polynomial that is irreducible mod $p$ (u can easily see that the your ideal is not maximal). So when an ideal of $\mathbb{Z}[X]$ is maximal, the quotient the finite field $\mathbb{Z}/p\mathbb{Z}$.