Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field I want to show that $ \mathbb Z[x] /<5,x^3+x+1>$
I am currently self-studying and it is a problem of previous graduate school entrance exam.
 I studied abstract algebra through Fraleigh's book, but it seems that this book does not contain information about ideal generated by two elements like $<5,x+3+x+1>$ of $R[x]$ where $R$ is an integral domain with unity and condition under what those ideals are maximal.
 So basically, my knowledge is just


*

*the definition of an ideal generated by two element is and 

*$F[x]/ <f>$ is a field when $F$ is a field and $f$ is irreducible in $F[x]$.
So can someone explain how to solve this problem under my knowledge and recommend a text book or articles about dealing above material? Thanks in advance.  
 A: The first step is to show that
$$
\frac{{\Bbb Z}[x]}{(5,x^3+x+1)}\simeq\frac{{\Bbb F}_5[x]}{(x^3+x+1)}
$$
For this you can just define an obvious morphism and show that is injective and surjective ($\Bbb F_q$ denotes the finite field with $q$ elements).
Next, you need to show that $x^3+x+1$ is irreducible in ${\Bbb F}_5[x]$. For, note that any decomposition $x^3+x+1=f(x)g(x)$ would give rise to a factor of degree $1$ and polynomials of degree $1$ have roots. Thus, to show irreducibility it is enough to show that  $x^3+x+1$ has no roots in $\Bbb F_5$.
I leave the details to you.
A: Let us first see how will be the elements of $\frac{{\Bbb Z}[x]}{\langle 5, x^3+x+1\rangle}$ look like.$$ \frac{{\Bbb Z}[x]}{\langle 5,x^3+x+1\rangle} =\Big\{p(x)+5q(x)+r(x)(x^3+x+1):\ p(x),q(x),r(x)\in \Bbb Z[x] \Big\}.$$ And there is a natural homomorphism between $\frac{{\Bbb Z}[x]}{\langle 5,x^3+x+1\rangle}$ and $\frac{{\Bbb Z_5}[x]}{\langle x^3+x+1\rangle}$ defined as  $$\phi\ :\ \frac{{\Bbb Z}[x]}{\langle 5,x^3+x+1\rangle} \to \frac{{\Bbb Z_5}[x]}{\langle x^3+x+1\rangle}$$ by, $$ p(x)+5q(x)+r(x)(x^3+x+1)\mapsto \overline{p(x)}+\langle x^3+x+1 \rangle. $$ where $\overline{p(x)}$ shows that the coefficients are modulo 5. Clearly, this map is an isomorphism (Easy to verify). So now it is enough to prove  that $\frac{{\Bbb Z}[x]}{\langle x^3+x+1\rangle}$ is a field $\iff \langle x^3+x+1\rangle$ is a maximal ideal $\iff \langle x^3+x+1\rangle $ is irreducible over $\Bbb Z_5.$ Since $x^3+x+1$ is irreducible iff it has a root in $\Bbb Z_5[x]$. Since it does not have a root in $\Bbb Z_5[x] \implies $ it is irreducible in $\Bbb Z_5[x]$ and hence,  $\frac{{\Bbb Z_5}[x]}{\langle x^3+x+1\rangle}$ is a filed so $\frac{{\Bbb Z}[x]}{\langle 5, x^3+x+1\rangle}$ is also a field.
