Definition:Field of Rational Numbers

Definition

The field of rational numbers $\struct {\Q, + \times, \le}$ is the set of rational numbers under the two operations of addition and multiplication, totally ordered by $\le$.

When the totally ordering $\le$ is subordinate or irrelevant in the context in which it is used, $\struct {\Q, + \times}$ is usually seen.

Also see

Thus:

$\struct {\Q, +}$ is the additive group of rational numbers
$\struct {\Q_{\ne 0}, \times}$ is the multiplicative group of rational numbers
The zero of $\struct {\Q, + \times}$ is $0$
The unity of $\struct {\Q, + \times}$ is $1$.