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, with an ordering $\le$ compatible with the ring structure of $\Q$.

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

Also see

• Rational Numbers form Ordered Field

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 $1$.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $2$
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts