Question

a) Show that$\raisebox{1ex}{$\overline{){\mathbf{\mathbb{Z}}}_{\mathbf{2}}\left[x\right]}$}\!\left/ \!\raisebox{-1ex}{$\left(x^3+x+1\right)$}\right.$is a field.

b) Show that the field$\raisebox{1ex}{${\mathbf{\mathbb{Z}}}_{\mathbf{2}}\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^3+x+1\right)$}\right.$contains all three roots of${\mathbf{x}}^{\mathbf{3}}\mathbf{+}\mathbf{x}\mathbf{+}\mathbf{1}$.

Short answer

b) It is proved that$\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^3+x+1\right)$}\right.$ contains all three roots of $x^3+x+1$.

Step-by-step solution

Theorem 5.11

Theorem 5.11states that, for a field $F$and irreducible polynomial $p\left(x\right)$in $F\left[x\right]$, $\raisebox{1ex}{$F\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(p\left(x\right)\right)$}\right.$ will be an extension field $K$of $F$, which contains a root of $f\left(x\right)$.

Show that ℤ2[x](x3+x+1) contains all three roots of x3+x+1

b)

It is known that $\left[x\right]\in \raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^3+x+1\right)$}\right.$willbe a root of$y^3+y+1$ (Theorem 5.11).

There will be${\left(x^2\right)}^3+\left(x^2\right)+1={\left(x^3+x+1\right)}^2$ in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$. As a result,$\left[x^2\right]$ is a root of $y^3+y+1$.

Moreover, ${\left(x^2+x\right)}^3+\left(x^2+x\right)+1=\left(x^3+x+1\right)\left(x^3+x^2+1\right)$. Therefore,$\left[x^2+x\right]$ also willbe a root of $y^3+y+1$.

Hence, it is proved that $\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^3+x+1\right)$}\right.$contains all three roots of $x^3+x+1$.