Rational Numbers form Totally Ordered Field

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Theorem

The set of rational numbers $\Q$ forms a totally ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$.


Proof

Recall that by Integers form Ordered Integral Domain, $\struct {\Z, +, \times, \le}$ is an ordered integral domain

By Rational Numbers form Field, $\struct {\Q, +, \times}$ is a field.

In the formal definition of rational numbers, $\struct {\Q, +, \times}$ is the quotient field of $\struct {\Z, +, \times, \le}$

By Total Ordering on Quotient Field is Unique, it follows that $\struct {\Q, +, \times}$ has a unique total ordering on it that is compatible with $\le$ on $\Z$.

Thus $\struct {\Q, +, \times, \le}$ is a totally ordered field.

$\blacksquare$


Sources