Field with four elements If $F=\{0,1,a,b\}$ is a field (where the four elements are distinct), then:

1.What is the characteristic of $F$?

2.Write $b$ in terms of the other elements.

3.What are the multiplication and addition tableau of these operations?
The addition: $\;a+b=0\implies a=-b=b\;$ , since $\;\text{char}\, F=2\;$ . It also can't be $\;a+b=a\;,\;\;a+b=b\;$ , else $\;b=0\;$ or $\;a=0\;$ . Thus it must be $$\;a+b=1\implies b=1-a=1+a\;$$

Generalize the above and get the addition table.

As for multiplication $\;ab\neq 0,a,b\;$ , else $\;a=0\;$ or $\;b=0\;$ , or $\;a,b=1\;$ and none of this is true, thus it must be $\;ab=1\iff b=a^{-1}\;$ . Generalize now for the table.
Hint : What can u get of this $$F=\frac {Z_2[x]}{&lt;x^2+x+1&gt;}$$