Addition over a field

Question

I'm looking at $\mathbb{F}_4 = \{0,1,\omega,\omega^2\}$ and wondering how to do addition in this field.

I have three expressions: $16$, $4\omega + 4\omega^2$, $6 + 5\omega + 5\omega^2$

I understand how to do addition over $\mathbb{Z}_4$, since it's just modular arithmetic, but I believe it is not the same over $\mathbb{F}.$

Answer 1

Hint In this field, we have $0 = 1 + 1 + 1 + 1 = (1 + 1) (1 + 1)$, so we must have $1 + 1 = 0$. For the same reason, $\omega + \omega = 0$ (and $\omega^2 + \omega^2 = 0$).

Among the remaining sums to determine is $1 + \omega$. Can this sum be any of $0, 1, \omega$?