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?

For (1), suppose that the characteristic is $4$. Then, $1 + 1 \neq 0$, but $$(1 + 1)(1 + 1) = 1 + 1 + 1 + 1 = 0,$$ so $1 + 1$ is a zero divisor, a contradiction, and hence the characteristic is $2$. One can use this sort of argument to show that the characteristic of any integral domain is either $0$ or prime.
Since the characteristic is $2$, for every $x \in F$ we have $x + x = (1 + 1)x = 0$, so every element has order $\leq 2$ under $+$, and this determines up to isomorphism the underlying group $(F, +)$, which in particular lets you solve (2) and the first part of (3). The second part of (3) follows quickly from this.
