# Monomorphism from Rational Numbers to Totally Ordered Field

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## Theorem

Let $\struct {F, +, \circ, \le}$ be a totally ordered field.

There is one and only one (ring) monomorphism from the totally ordered field $\Q$ onto $F$.

Its image is the prime subfield of $F$.

## Proof

Follows from:

- Characteristic of Ordered Integral Domain is Zero
- Order Embedding between Quotient Fields is Unique.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.11$