What is the general property a field in which addition can be "undone" by addition

Question

For example, in GF(2) we can do the following:

p = 110
k = 010
c = p+k = 110+010 = 100

If we want to calculate p or k from c and one of the other numbers, we can simply add:

c+p = k
100+110 = 010

c+k = p
100+010 = 110

What is this property called, why does it work for GF(2), and what other fields does this work for?

Answer 1

If you want both $c+k=p$ and $c+p=k$, then you get $2(p-k)=0$. If $p$ and $k$ have to be generic elements of the field, you end up asking that $2a=0$ for every $a$ in the field, i.e. that it has characteristic $2$.

On the other hand, if you have a field of characteristic $2$, then $a=-a$ for every element and so if $a=b+c$ then $b=a-c=a+c$ and $c=a-b=a+b$.

Answer 2

This works in a ring of characteristic 2, meaning $1+1=0$. In such a ring $x=-x$ for all $x$, so $(y+x)+x=y$ for all $x,y$.