Why is $1 + 1 = 0$ in $\{0, 1\}$ (binary field) and not 1 or 2?

Question

Stuck on the simplest case in my foray into fields...

I know there is a really similar question out there, but I can't find any contradiction with the field axioms if 1 + 1 = 1 instead of 0.

Can someone explicitly show me the logic behind why 1 + 1 = 0 over the binary field and not 1 or 2?

Answer 1

Well 2=0 in the binary field. Also, a field is an (abelian) group under addition so it satisfies cancellation: $a+b = a+c \Leftrightarrow b =c$. Since $0$ is stipulated to be the additive identity we have

$$1 + 1 = 1 = 1+0 \Leftrightarrow 1= 0$$

But we know $1 \neq 0$ , so $1 + 1 \neq 1$ in any field. This is a general application of the fact that in any group

$$a^2 = a \Leftrightarrow a = e$$.

Answer 2

What's the additive inverse of $1$ if $1+1=1$?

The field axioms unambiguously decide what $+$ and $\cdot$ are for $\mathbb{F}_2$, and the above counter-question should give you an idea why.