abstract algebra - How the following multiplication table is solved ( related to $F_2[X]/f(x)$ )

$F_2$ is polynomial field of group of integer modulo $2.f(x)$ is $x^2 + x + 1$.

I didn't got how the multiplication is happening in the table.I referred to many sources related to this topic but still i am facing difficulty in understanding it.I will be very thankful if someone explains the concept behind it.

Answer

This is the multiplication table for the field ${\Bbb F}_4 = {\Bbb F}_2[x]/\langle x^2+x+1\rangle$ consisting of the residue classes of the elements $0,1,x,x+1$ which are the remainders of ${\Bbb F}_2[x]$ modulo $x^2+x+1$.

For instance, $[x] \cdot [x+1] = [x\cdot(x+1)] = [x^2+x]$ and the residue class of $x^2+x$ modulo $x^2+x+1$ is $[1]$, i.e., $x^2+x = 1\cdot (x^2+x+1) + 1$ with quotient $q(x)=1$ and remainder $r(x)=1$. This is an elementary way to view this field extension.