i.e 01. It has to abide by the other 9 axioms of a field.
If you don't stipulate that 0 ≠ 1, then, yes, there is such a field. It has exactly one element, namely the additive/multiplicative identity. However, people generally ignore this field when talking about fields, as it doesn't have much structure. So people usually stick in an extra condition that 0 ≠ 1.
