Can someone help me make a multiplication and addition table for a field $\{0, 1, a, b, c\}$.
I understand the basic rows and columns of $x+0$, $x\cdot 0$, $x\cdot 1$ and that the element plus itself is 0 (so $x+x=0$), but I don't understand how to do the rest. Through logical deduction and making the table symmetrical I think I have the rest but I can't prove it.
My attempt: $$ \begin{array}{c|c|c|c|c|c} + & 0 & 1 & a & b & c \\ \hline 0 & 0 & 1 & a & b & c \\ \hline 1 & 1 & 0 & c & a & b \\ \hline a & a & c & 0 & c & b \\ \hline b & b & a & c & 0 & a \\ \hline c & c & b & b & a & 0 \end{array} \quad \begin{array}{l|l|l|l|l|l} \cdot & 0 & 1 & a & b & c \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 1 & 0 & 1 & a & b & c \\ \hline a & 0 & a & b & c & 1 \\ \hline b & 0 & b & c & 1 & a \\ \hline c & 0 & c & 1 & a & 0 \end{array} $$