Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field

Question

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

1. the definition of an ideal generated by two element is and

2. $F[x]/ <f>$ 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.

Answer 1

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.

Answer 2

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.