# Definition of rational numbers

We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some other one. I know that we can define integers in ordered field so maybe this axiom should state that every number is quotient of two integers ? Or maybe we can choose some simpler one. So what axiom/axioms should we choose instead of completness axiom to define rational numbers ? Thanks in advance.

• See Rational numbers : "The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of $\mathbb Q$. The rational numbers are therefore the prime field for characteristic zero." – Mauro ALLEGRANZA Dec 2 '18 at 12:27

More precisely, let $$F$$ be an ordered field without subfields (except for $$F$$ itself). Then it is not hard to prove that
• $$\mathbb Q$$ is one such field;
• every such field is isomorphic to $$\mathbb Q$$.