Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point.
And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.