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.

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    $\begingroup$ 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." $\endgroup$ – Mauro ALLEGRANZA Dec 2 '18 at 12:27

We can define the field of rational numbers as the smallest ordered field.

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 $\mathbb Q$.

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