How to understand $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle $

Question

I read that $\mathbb{Z}_2$ is equal to $\{0,1\}$. On the other hand, is $\mathbb{Z}_2[x]$ is equal to the set of polynomials with coefficients $a_i \in \mathbb{Z}_2$ ?

What would then be the elements of $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle $? , which from the text am reading is supposed to mean the quotient set formed from $\mathbb{Z}_2[x]$ and the irreducible polynomial $x^3+x+1$ ... Specifically, $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle $ is a field that should contain $8$ elements, but I could not figure out what these elements are ... Any help ?

Answer 1

As @Wuestenfux notes, it might be best to read more about quotients.

But here is a simple intiutive explanation that doesn't require much background knowledge.

$\mathbb{Z}_2[X]$ denotes the polynomials where each co-efficient is $0$ or $1$.

Since $x^3 + x + 1$ is irreducible (i.e. it cannot be factorized into smaller polynomials with coefficients in $\mathbb{Z}_2$),

$\mathbb{Z}_2[X] / \left<x^3+x+1\right>$ can be thought as those polynomials in $\mathbb{Z}_2[X]$ with degree at most 2. This is exactly the set $\{0,1,x,1+x,x^2,1+x^2,x + x^2, 1+x+x^2\}$.

Now, if you consider the addition operation ($+$) where each component is added modulo 2, and the multiplication operation ($\times$) defined as polynomial multiplication modulo $x^3+x+1$, then

It turns out this is exactly the field with eight elements $\mathbb{F}_8$

Answer 2

Hint: With Euclidean division, every polynomial in $\Bbb Z_2[x]$ can be written in the form $$ p(x) = q(x)(x^3 + x + 1) + r(x), $$ where $r(x)$ is a polynomial with degree at most $2$.