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


*

*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?

*What type of elements are in $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$?

*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.

*

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

*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 $I+r_1=I+r_2 \iff r_1-r_2\in I$.

*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 $Z/2Z$
