Why does 0 not equal 1 in fields?

Question

I know fields must have a 0 and a 1 in them, and apparently 0 cannot equal 1. Why is that? What provable properties do we lose if we allow 0 = 1?

Answer 1

There is only one ring in which 1=0, namely the ring with one element, so the same is true of fields in which 1=0.  This "trivial ring" has the annoying property that 0 is a unit, i.e. division by zero works.  Consequently, if we allow this trivial ring we will have to say many things awkwardly, for example, "let a be a non-zero unit", rather than "let a be a unit". Similarly, many theorems about all rings would have the zero ring as an exception, and we would have to constantly say things like "let R be a non-trivial ring" rather than "let R be a ring".

So far as I know it is simply to avoid such awkward statements that people usually insist that 1 not equal 0 in a ring or field.

Answer 2

a*0 = a*(0+0) = a*0 + a*0      / substract a*0

a*0 -a*0 = a*0 +a*0 - a*0

0 = a*0

Therefore a*0 = 0, for any value of a.