Explicit description of quotient ring of $\mathbb{Z}[x]$ I am studying for a prelim and I stumbled on this problem:
Describe explicitly the elements in the quotient ring $\dfrac{\mathbb{Z}[x]}{(3,x^3-x+1)}$. First of all I don't see why the ideal $(3,x^3-x+1)$ is a maximal ideal in $\mathbb{Z}[x]$. If there is anyone who can help me with this will be greatly appreciated.

You can solve this problem by a two step process.  First, let $J = (3, x^3-x+1)$, and let $I = (3)$.  These are ideals of $\mathbb Z[x]$ with $I \subset J$.  
The third isomorphism theorem says that
$$\mathbb Z[x]/(3,x^3-x+1) = \mathbb Z[x]/J \cong \frac{\mathbb Z[x]/I}{J/I}$$
In other words, the ring you are looking for can be found by taking the ring $\mathbb Z[x]/I$ and modding out by an ideal therein.
Note that $\mathbb Z[x]/I \cong \mathbb F_3[x]$, where $\mathbb F_3$ is the field with three elements.  Inside this ring, $J/I$ is just the ideal in $\mathbb F_3[x]$ generated by $x^3-x+1$.  
The problem becomes to describe the elements of the quotient ring $\mathbb F_3[x]/(x^3-x+1)$.  To do this, you should first determine whether or not $x^3-x+1$ is irreducible in $\mathbb F_3[x]$.  
