Constructing a multiplication table for a finite field 
Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$ and let $F=\mathbb{Z}_2(\alpha)$, where $\alpha$ is a root of $f(x)$. Show that $F$ is a field and construct a multiplication table for $F$.

Can you please help me approach this problem? I've tried searching around, but I don't really know what I'm looking for!
Thanks.

Using repeatedly the relation $\,\alpha^3=\alpha+1$, knowing a basis of $\mathbf Z_2[\alpha]$ over $\mathbf Z_2$ is $\{1,\alpha,\alpha^2\}$ we have the following table of multiplication:

