Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$ Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?
That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?

Just to expand Asaf's answer, take polynomials in $x$, modulo $x^2+x+1$. The coefficients are in $\mathbb{Z}/2\mathbb{Z}$.  They look like linear polynomials in $x$.  That gives a field of four elements.  If you can find a cubic $C(x)$ which is nonzero $(\mod 2)$ for $x=0$ and $x=1$,  you can take polynomials modulo $C(x)$, which is all the quadratic polynomials, and is a field of eight elements.
