How can I find the sum and multiplication table of a $F_4$ field?

Question

From the field $\{0,1,A,B\}$ I know that, by exhaustion, $A\cdot B=1$ but how do I get $A \cdot A $ and $B\cdot B$? How can I get the sum table? We haven't been thought almost anything about fields in class yet so I am stuck even with the other answers on the internet. I need a full explanation. I don't know what cyclic means or anything pretty much about fields.

Answer 1

Note that $B=A+1$. Hence if you already know $AB=1$, we conclude $AA=A(B-1)=AB-A=1-A=1+A=B$ and $BB=(A+1)B=AB+B=1+B=2+A=A$.