I have this question which I don't know how to approach:
Let ${F}_{2} = {Z}/2Z$, find representatives for the residue classes of ${F}_{2}[X]$ modulo the polynomial $f(x)$ and compute the multipication table for the ring ${F}_{2}[X]/(f(x))$ where
- $f(x) = x+1$
- $f(x) = x^2+x+1$
- $f(x) = x^2+1$
which are those rings are fields?
I understand $F_2$ and ${F}_{2}[X]$ but I don't know what the residues are and how to compute the multiplication table or how multiplication is defined in this ring. I suppose that checking if something is a field is easy once you have the multiplications table.