Counting elements of $\Bbb{Z}/2\Bbb{Z}(\alpha)$

Question

I have the field $K=\Bbb{Z}/2\Bbb{Z}$, I proved that the polynomial $P(X)=X^3+X^2+1$ is irreducible. Then I know that the quotient $K[X]/P$ is a field of $8$ elements. Let now $\alpha$ be a root in an extension of $K$, as $\alpha$ is algrebaic, we have $K[\alpha]$ isomorphic to $K(\alpha)$.

It's asking to enumerate elements of $K(\alpha)$

How can I do that ?

Answer 1

Hint:

Enumerate all polynomials of degree less than $3$ with coefficients in $\mathbf Z/2\mathbf Z$.