This question already has an answer here:
- Is it possible to make integers a field? 3 answers
I think that these two sets cannot be turned into fields by re-defining addition or multiplication (or both) but I am not sure how to prove this only from axioms of the field and (if needed) some elementary properties of fields that follow directly from axioms of the field.
My intuition is that $\mathbb N_0$ and $\mathbb Z$ are not "dense enough", as is, for example, $\mathbb Q$, to be able to become fields.
How would you prove that we can(or cannot) turn them into fields?