# Definition:Field of Rational Numbers

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## 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

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