Questions about the quotient ring $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$

Question

I'm just starting to learn about quotient rings. I was able to think about what type of of elements are generated by some $a$ when $\langle a \rangle$ is simply an integer, and also with simple quotient rings, but I just don't get this example.

My question about the quotient ring given, $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$ are:

1. What kind of elements are in $\langle x^2+x+1 \rangle$? Applying the definition, I get something like $\{(x^2+x+1)q(x) \colon q(x) \in (\mathbb{Z}/2\mathbb{Z})[x]$}. I cannot readily list out the elements though. What is an intuitive way of thinking about the principal ideal generated by a polynomial?
2. What type of elements are in $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$?
3. The author claims $x^2 = x+1$. I don't see why.

I am self learning this, so this should help illuminate how I may think about quotient rings as I proceed. Thanks in advance.

Answer 1

The golden rule:

In a quotient ring $R/I$ a residue class $\hat a$ is zero iff $a\in I$.

So (the residue class of) $x^2+x+1$ is $0$ in $(\mathbb Z/2\mathbb Z)[x]/(x^2+x+1)$. By an abuse of notation we can write $x^2+x+1=0$ (instead of $\widehat{x^2+x+1}=\widehat{0}$), and since in $\mathbb Z/2\mathbb Z$ we have $1=-1$ then $x^2=x+1$.

In order to find the form of the elements of $(\mathbb Z/2\mathbb Z)[x]/(x^2+x+1)$ start with a polynomial $f\in(\mathbb Z/2\mathbb Z)[x]$ and write $f(x)=(x^2+x+1)g(x)+r(x)$ with $\deg r\le1$. Now, by taking the residue classes modulo the ideal $(x^2+x+1)$ we get $f(x)=r(x)$, so the elements of $(\mathbb Z/2\mathbb Z)[x]/(x^2+x+1)$ are of the form $ax+b$ with $a,b\in\mathbb Z/2\mathbb Z$. (These are in fact only four: $0$, $1$, $x$, $x+1$.)

Answer 2

1. The set you're written is correct. In any commutative ring ideal generated by one element $a$ is just $<a>=\{ra|r\in R\}$

2. Ideal and quotient ring is similar to normal subgroup and quotient group , the element in $R/I$ (here your $R$ is $Z/2Z[x]$ and $I$ is $<x^2+x+1>$) is of the form $I+r$ with $r\in R$ and $I+r_1=I+r_2 \iff r_1-r_2\in I$.

3. It's a short hand notation of $<x^2+x+1>+x^2=<x^2+x+1>+x+1$, which is equivalent to $x^2-x-1\in <x^2+x+1>$. In fact $x^2-x-1=x^2+x+1$ because the coefficient is in $Z/2Z$