Let $\mathbb{Z}_2=F$. Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$. Suppose $a$ is a zero of $f(x)$ in some extension field of $\mathbb{Z}_2$. How many elements does $F(a)$ have and express each member of $F(a)$ in terms of $a$.
My Try:
Some info I understand:
$f(x)$ has degree $3$ and doesn't have a root in $\mathbb{Z}_2$, so we conclude that because the degree is low, there are $f(x)$ is irreducible in $\mathbb{Z}_2$. Since $f(x)$ is irreducible I know that $\mathbb{Z}_2[x] / \langle f(x)\rangle $ is a field(I'm not sure if this information is relevant to the question or not.)
$a$ is a root so $f(a)=0$ in some extension field of $\mathbb{Z}_2$
Can someone help push me in the right direction?
The field consists of all polynomials in $\mathbb{Z}_2[x]$ of degree at most $2$ evaluated at $a$.
This is because the field $\mathbb{Z}_2[x]/\langle f(x) \rangle$ is "produced" by taking all polynomials in $\mathbb{Z}_2[x]$ and reduce them modulo the polynomial $f(x)$.