[Math] Is the quotient ring $\mathbb Z[x]/(5, x^3+x+1)$ a field

Question

The problem is this:

Show that $A=\mathbb Z[x]/(5, x^3+x+1)$ is a field.

1. I tried to show that that ideal is a maximal ideal, but failed.

2. Since A is finite set, so it suffices to show that (5, x^3+x+1) is a prime ideal (because finite integral domain is field), but failed.

3. I tried elementary proof and I succeeded but this is too complicated.

I don't know how to solve this problem. Is there any good solution?

p.s. I'm a undergraduate student. So please use only undergraduate algebra.

I'm not good at english. So please understand me if there are grammatical error.

Answer 1

Note that $A$ is isomorphic to $\mathbb{Z}/(5)[x]/(x^3+x+1)$. That means that it is enough to check that $x^3+x+1$ is irreducible modulo $5$, which is easy. You only have to insert $5$ different values for $x$.