I have an polynomial $x^4+x+1 \in \mathbb{Z}\left\{ x\right\}$ and I want to construct an extension field of $\mathbb{Z}_2$ that include the roots of that polynomial. So is this the right approach?
Let E be the extension field. $$E= \mathbb{Z}_2 / <x^4+x+1> $$?
If so, how do I find the root of this polynomial? And what is the range of the extension field?
On
Only degree $2$ irreducible polynomial over $\mathbb{Z}_2$ is $x^2+x+1$. But it's square is $x^4+x^2+1\neq x^4+x+1$. Also note that $x^4+x+1$ has no roots. Therefore $x^4+x+1$ has no irreducible factor of degree $1$ or $2$ and hence it must be irreducible. This shows that $E$ is a field.
Now consider the polynomial ring with coefficients in $E$, i.e. $E[y]$, and the polynomial $f = y^4+y+1$. Insert $x$ to $f$. We have $f(x)=x^4+x+1=0$. Hence it has roots in $E$.
Edit : It is a better notation to use $x+I$ instead of $x$, but I don't think it leads to a confusion.
The field $\;\Bbb F:=\Bbb F_2[x]/I\;,\;\;\text{with}\;\;I:=\langle x^4+x+1\rangle\;$, (it is a because $\;x^4+x+1\;$ is irreducible), contains the element $\;\omega:=x+I\;$ . Check this element is a root of the quartic.
In general, if $\;\Bbb K\;$ is a field and $\;f(x)\in\Bbb K[x]\;$ is irreducible, we have that
$$\dim_{\Bbb K}\Bbb K[x]/\langle f(x)\rangle =\deg f$$
and thus your extension's degree is $\;4\;$ .