Addition and Multiplication in $F_4$

$\Bbb{F}_4$ is the finite field of order $4$. It is not the same as $\Bbb{Z}_4$, the integers modulo 4. In fact, $\Bbb{Z}_4$ is not a field. $\Bbb{F}_4$ is the splitting field over $\Bbb{F}_2 = \Bbb{Z}_2$ of the polynomial $X^4 - X$. You get the addition table by observing that $\Bbb{F}_4$ is a 2-dimensional vector space over $\Bbb{F}_2$ with basis $1$ and $x$ where $x$ is either of the roots of $X^4 - X = X(X - 1)(X^2 + X + 1)$ that is not in $\Bbb{F}_2$. You get the multiplication table by using $x^2 + x + 1= 0$ to simplify the expressions for the products of $x$ and $x + 1$.

The addition and multiplication tables are essentially forced on us. Suppose we have a field with four elements. Under addition it is an abelian group. We must have the elements zero and one. The first question is what is the additive order of one? It must be a divisor of four, so it is either four or two. In the first case, we have that $\;\{0,1,2:=1+1,3:=2+1,4:=3+1\}\;$ but there are only four elements so we must have $\,4=0\,$ since it can't equal any of the three other elements. Now we have that $\;2\times 2=4=0,\;$ so zero divisors exist and thus it can't be a field.

The only other possibility is that $\;1+1=0\;$ which implies that there are two other elements. Call one of them $\;x\;$ and the other is $\;x+1\;$ since it can't equal any of the other three elements. In the same way, $\;x+x=0\;$ since it can't equal any of the other three elements. This is enough to uniquely determine the sixteen entries in the addition table.

For multiplication, the characteristic properties of zero and one determine twelve of the entries in the multiplication table. We get $\;x\times x = 1+x\;$ since it can't equal any of the other three elements. We also get $\;x \times(1+x) = 1\;$ since the multiplicative inverse of $\;x\;$ can't equal any of the other three elements. Finally, $\;(1+x)\times(1+x)=x\;$ and we are done.