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 $24.11$