GF(3) can be constructed as follows by polynomial $p(x)=x+1$:
$0=0$
$1=1$
$\alpha=2$
GF(5) can be constructed as follows by polynomial $p(x)=x+2$:
$0=0$
$1=1$
$\alpha=3$
$\alpha^2=\alpha\cdot\alpha=3\cdot3\bmod5=4$
$\alpha^3=\alpha^2\cdot\alpha=4\cdot3\bmod5=2$
Question is, what can be defined as element $\alpha$ in GF(2) and what primitive polynomial can be used to construct this field?