abstract algebra - Addition in finite fields

For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of degree 3, and creating the splitting field for that polynomial.

The polynomial I chose for the field of order 8 is $x^3 + x + 1$. Since every finite field's multiplicative group is cyclic, it's my understanding that I can write the 6 elements of this field that are not 0 and 1 and $a, a^2, ..., a^6$, where $a^7 = 1$. And if my thinking is correct about that, I know how multiplication works. But I'm lost on how to develop addition in this field. I know in this case that since 1 + 1 = 0, every element should be its own additive inverse, but beyond that I'm lost as to how, for example, I should come up with a proper answer for what $1 + a$ is.

That is, assuming I'm right about the first parts.

An answer that could help me understand how to do this in general would be very helpful, as I need to do this for a field of another order as well.

Answer

Hint: $a$ being a root of your polynomial gives you the relation $a^3+a+1=0$