Hi all, I am going through a Algebra textbook and I need a bit of help with this question, so thank you in advance for any help.

Question: Prove that the set of symbols $\displaystyle \{a+bi \ | \ a,b \in F_3 \}$ forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers. Will the same method work for $\displaystyle F_5$? For $\displaystyle F_7$? Explain.

So I can see that the nine elements are: $\displaystyle \{0,1,2,i,1+i,2+i,2i,1+2i,2+2i \}$ but I am confused as to where to go from here.

June