# Integers form Subdomain of Rationals

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## Theorem

The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.

## Proof

The rational numbers are defined as the quotient field of the integers.

From its method of construction, it follows that the integers $\Z$ are a subset of the rational numbers $\Q$.

Hence the result, from the definition of subdomain.

$\blacksquare$

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.1$: Subrings: Examples $1$