Do diagonal elements in the Galois Field addition tables have to be zeros?

Question

I've seen addition and multiplication tables for Galois Fields, where the addition table is simply modular arithmetic, and some tables where the diagonal elements are zeros (i.e. the additive inverse of an element is itself). From what I can see, both types of tables fit the definition of a field. So, are both these tables correct?

Answer 1

No, diagonal elements of Galois fields do not have to be $0$. They can be, though. If they are, then by definition, this means that the characteristic of the field is $2$, so the order of the field is some power of $2$. For a Galois field with odd characteristic, $0+0$ will be the only $0$ in the diagonal of the addition table. That's because if $a+a = 0$, and the characteristic of the field is not $2$, we have $$ 0 = a+a = 2a $$ which, since $2\neq 0$, means that $a = 0$.