Sanity check the description of $\Bbb{Z}_2[x]/\langle x^3+x+1\rangle$.

Question

This question originates from Pinter's Abstract Algebra, Chapter 27, Exercise C5.

Describe $\Bbb{Z}_2[x]/\langle x^3+x+1\rangle$, as in part 4.

Part 4: Explains why there are exactly four elements in $\Bbb{Z}_2[x]/\langle x^2 + x + 1\rangle$, lists the four elements, and gives their addition and multiplication tables.

Let $p(x)=x^3+x+1$. The polynomial $p(x)$ is irreducible in $\Bbb{Z}_2[x]$ as it has no root in $\Bbb{Z}_2$.

1. There is exactly a total of eight elements in $\Bbb{Z}_2[x]/\langle x^3+x+1\rangle$, as that's the number of all the possible polynomials with degree less than the degree of $p(x)$ in $\Bbb{Z}_2[x]$.
2. Specifically, these eight elements $\{0,1,x,x^2,1+x,1+x^2,x+x^2,1+x+x^2\}$ form a field which is isomorphic to the extended field $\Bbb{Z}_2(c)$, where $c$ is a root of $p(x)$.
3. Note in particular $c^3+c+1=0$. Hence $c^3=-c-1=c+1$ in $\Bbb{Z}_2$.
4. The addition and multiplication tables of these eight elements are given below.

Addition table: \begin{array}{ c| c c c c c c c c } + & 0 & 1 & x & x^2 & 1+x & 1+x^2 & x+x^2 & 1+x+x^2 \\ \hline 0 & 0 & 1 & x & x^2 & 1+x & 1+x^2 & x+x^2 & 1+x+x^2 \\ 1 & 1 & 0 & 1+x & 1+x^2 & x & x^2 & 1+x+x^2 & x+x^2 \\ x & x & 1+x & 0 & x+x^2 & 1 & 1+x+x^2 & x^2 & 1+x^2 \\ x^2 & x^2 & 1+x^2 & x+x^2 & 0 & 1+x+x^2 & 1 & x & 1+x \\ 1+x & 1+x & x & 1 & 1+x+x^2 & 0 & x+x^2 & 1+x^2 & x^2 \\ 1+x^2 & 1+x^2 & x^2 & 1+x+x^2 & 1 & x+x^2 & 0 & 1+x & x \\ x+x^2 & x+x^2 & 1+x+x^2 & x^2 & x & 1+x^2 & 1+x & 0 & 1 \\ 1+x+x^2 & 1+x+x^2 & x+x^2 & 1+x^2 & 1+x & x^2 & x & 1 & 0 \end{array}

Multiplication table:

\begin{array}{ c | c c c c c c c c } * & 0 & 1 & x & x^2 & 1+x & 1+x^2 & x+x^2 & 1+x+x^2 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & x & x^2 & 1+x & 1+x^2 & x+x^2 & 1+x+x^2 \\ x & 0 & x & x^2 & 1+x & x+x^2 & 1 & 1+x+x^2 & 1+x^2 \\ x^2 & 0 & x^2 & 1+x & x+x^2 & 1+x+x^2 & x & 1+x^2 & 1 \\ 1+x & 0 & 1+x & x+x^2 & 1+x+x^2 & 1+x^2 & x^2 & 1 & x \\ 1+x^2 & 0 & 1+x^2 & 1 & x & x^2 & 1+x+x^2 & 1+x & x+x^2 \\ x+x^2 & 0 & x+x^2 & 1+x+x^2 & 1+x^2 & 1 & 1+x & x & x^2 \\ 1+x+x^2 & 0 & 1+x+x^2 & 1+x^2 & 1 & x & x+x^2 & x^2 & 1+x \end{array}

Is this correct?

Answer 1

As Chris Custer said, this is just a $3$-dimensional vector space with basis $\{1,x,x^2\}$, so yes, there are $2^3=8$ elements that you listed.

Addition is just the usual componentwise addition (mod $2$, of course).

For multiplication, you can make your life easier than doing the whole table. Since $x^3 = -x-1 = x + 1$ and $x^4 = x^3\cdot x = x^2+x$, we have

\begin{align} \left(a+b x+c x^2\right) & \left(d+e x+f x^2\right)\\ & =a d + (b d + a e) x + (c d + b e + a f) x^2 + (c e + b f) x^3 + c f x^4\\ &= a d + c e + b f + (b d + a e + c e + b f + c f) x + (c d + b e + a f + c f) x^2. \end{align}

Answer 2

This looks right. It's a $3$-dimensional $\Bbb Z_2$ vector space.