Suppose we have a field F = {0, 1}
1 + 1 = ?
How does one determine if 1+1 = 1 or if 1+1 = 0?
Can I define it to be one of the two choices?
Another way of saying the same thing is that every element of a field must have an additive inverse. The additive inverse of 1 cannot be 0 because $\displaystyle 1+ 0= 1\ne 0$. Since 0 and 1 are the only members of the field, 1 must be its own inverse: 1+ 1= 0.