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?

A field has characteristic a prime number $p$ or $0$, (consider the homomorphism $\mathbf Z\rightarrow F, \enspace n\mapsto n\cdot 1$.
Any finite field $\mathbf F_{p^n}$ is an extension of degree $n$ of the prime field $\mathbf F_p$ and it is a simple extension of $\mathbf F_p$  generated  by a root of any irreducible polynomial of degree $n$ in $\mathbf F_p[x]$. In the present case the only irreducible polynomial of degree $2$ is $x^2+x+1$. Let $\omega$ be one of its roots; the other root is its inverse, $\omega+1$.
Here is its multiplication table:
$$\begin{array}{c|cccc}
&0&1 &\omega&\omega+1\\\hline
0&0&0&0&0\\\hline
1&0&1 &\omega&\omega+1\\\hline
\omega&0&\omega&\omega+1&1\\\hline
\omega+1&0&\omega+1&1&\omega
\end{array}$$
