Galois Field of order 4 operation with addition

Before posting any answers to my question; just a little note that, I only know the name of this such thing but have no clue what it is, i.e. the Galois Field of order 4.

Also, I have looked at the other posts that might explain it but it doesn't really clear it up for me, because how I go about these things is by showing that it fails one of the axioms of a field by the use of corollaries and theorems.

Suppose that we have a finite-field $F = \{0,1,a,b\}$.

Question: How does $1+1 = 0$?
I tried to think through in my head the other options and show that it can't be those other options, i.e. $1+1 \neq 1$ was the easy one.

$1+1 = 1 \implies 1 = 0$ but $1 \neq 0$ thus a contradiction.

I'm not sure how to go about how to show that $1+1 \neq x$?

Thanks.