In the definition of a field one of the required properties is that
every element other than zero has a multiplicative inverse.
It's vague whether the zero is forced not to have an inverse or not, as in is the property
every element other than zero has a multiplicative inverse and zero does not
or
every element other than zero has a multiplicative inverse and zero may or may not?
Regardless, what differences would occur if we adopted one definition as opposed to the other?
Best Answer
If you kept all of the other usual axioms, then
$$ 0 = 0 \cdot 0^{-1} = 1 $$
$$ x = x \cdot 1 = x \cdot 0 = 0 $$
and so all numbers are zero. This is rather degenerate. This is called the "zero ring"; usually we do not consider this ring to be a field.