How do i calculate a multiplication table for GF(8)?

Question

Could you please provide the steps involved in calculating a multiplication table for GF(8)?

Answer 1

Take an irriducibile polynomial of degree $3$ over the field $\Bbb F_2$ with two elements, e.g. $P(X)=X^3+X+1$. Then you know that $$ \Bbb F_8=\Bbb F_2[X]/(P(X)) $$ and that its 8 elements are represented by the 8 polynomials of degree $\leq2$. Thus you can construct the multiplication of $\Bbb F_8$ simply by multiplying these polynomials and taking the result modulo $P(X)$.