Question

Find a splitting field of ${\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathit{x}\mathbf{+}\mathbf{1}$ over${\mathit{Z}}_{\mathbf{2}}$ .

Short answer

$\frac{Z_2\left(x\right)}{(x^3+x+1)}$is the splitting field of $x^3+x+1$with $\frac{Z_2\left(x\right)}{(x^3+x+1)}=\{0,1,\alpha ,\alpha +1,{\alpha }^2,{\alpha }^2+1,{\alpha }^2+\alpha ,{\alpha }^2+\alpha +1\}$

Step-by-step solution

Definition of splitting field

A splitting field of a polynomial with coefficient in the smallest field over which the polynomial split or split into linear factors.

Find the splittingfield of  x3+x+1 

Let$Z_2=[0,1]$. Given polynomial$x^3+x+1$has no zero in$Z_2$as

$0^2+0+1\ne 0$

And

$1^2+1+1\ne 0$

Soby lemma given. Let$F$be a field and let$f\left(x\right)\in F\left[x\right]$be a polynomial of degree 2 or 3. Then$f\left(x\right)$is irreducible in$F\left[x\right]$if and only if$f\left(x\right)$has no root in$F$.

Therefore given polynomial$x^3+x+1$is irreducible over$Z_2$. Thus thesplitting field of$x^3+x+1$irreducible over$Z_2$is

$\frac{Z_2\left(x\right)}{(x^3+x+1)}=\{0,1,\alpha ,\alpha +1,{\alpha }^2,{\alpha }^2+1,{\alpha }^2+\alpha ,{\alpha }^2+\alpha +1\}$