The unique embedding of ℚ into an ordered field

Define $\iota$ on the integers by $\iota(n) = n \cdot 1_F$ for $n \ge 0$ and $\iota(-n) = -(n \cdot 1_F)$; by [L2] this is additive and multiplicative on $\mathbb{Z}$ and sends $1 \mapsto 1_F$.

For a rational $x = p/q$ with $q \ge 1$ define $\iota(x) = \iota(p)\,(q \cdot 1_F)^{-1}$, which makes sense because $q \cdot 1_F > 0 \ne 0$ has an inverse.

Well-defined: if $p/q = p'/q'$ with $q, q' \ge 1$, then $p q' = p' q$ in $\mathbb{Z}$, so [L2] gives $\iota(p)(q' \cdot 1_F) = \iota(p')(q \cdot 1_F)$, and multiplying by the positive $(q \cdot 1_F)^{-1}(q' \cdot 1_F)^{-1}$ yields $\iota(p)(q \cdot 1_F)^{-1} = \iota(p')(q' \cdot 1_F)^{-1}$; thus $\iota(x)$ is independent of the representative.

Multiplicativity: for $x = p/q$, $y = r/s$ one has $xy = (pr)/(qs)$, and $\iota(xy) = \iota(pr)((qs) \cdot 1_F)^{-1} = \iota(p)\iota(r)(q \cdot 1_F)^{-1}(s \cdot 1_F)^{-1} = \iota(x)\iota(y)$, using $(mn) \cdot 1_F = (m \cdot 1_F)(n \cdot 1_F)$ and $(uv)^{-1} = u^{-1} v^{-1}$.

Additivity: with $x + y = (ps + rq)/(qs)$, $\iota(x+y) = (\iota(p)(s \cdot 1_F) + \iota(r)(q \cdot 1_F))(q \cdot 1_F)^{-1}(s \cdot 1_F)^{-1} = \iota(p)(q \cdot 1_F)^{-1} + \iota(r)(s \cdot 1_F)^{-1} = \iota(x) + \iota(y)$, using the additive and multiplicative identities of [L2].

Positivity: if $x = p/q > 0$ in $\mathbb{Q}$ with $q \ge 1$, then $p \ge 1$ by [L1], so $\iota(p) = p \cdot 1_F > 0$ and $q \cdot 1_F > 0$ by [L2], whence $(q \cdot 1_F)^{-1} > 0$ by [L3] and $\iota(x) = \iota(p)(q \cdot 1_F)^{-1} > 0$ by [L4].

Uniqueness on $\mathbb{Z}$: let $\psi : \mathbb{Q} \to F$ be any field homomorphism; then $\psi(1) = 1_F$, additivity forces $\psi(n) = n \cdot 1_F = \iota(n)$ for $n \ge 1$, and $\psi(0) = 0$, $\psi(-n) = -(n \cdot 1_F)$, so $\psi = \iota$ on $\mathbb{Z}$.

Unit: $\iota(1) = \iota(1/1) = \iota(1)(1 \cdot 1_F)^{-1} = 1_F$; hence $\iota$ is a field homomorphism $\mathbb{Q} \to F$.

Order: for $x < y$ in $\mathbb{Q}$ we have $y - x > 0$, so $\iota(y) - \iota(x) = \iota(y - x) > 0$ by 2.3 and 2.4, that is $\iota(x) < \iota(y)$; thus $\iota$ is order-preserving.

Injectivity: if $x \ne y$ then $x < y$ or $y < x$, and 3.2 forces $\iota(x) \ne \iota(y)$; so $\iota$ is injective, an embedding of ordered fields.

Uniqueness on $\mathbb{Q}$: for $p/q \in \mathbb{Q}$, $\psi(p/q) = \psi(p)\psi(q)^{-1} = \iota(p)(q \cdot 1_F)^{-1} = \iota(p/q)$ since $\psi$ preserves products and inverses; hence $\psi = \iota$, so $\iota$ is the unique field homomorphism $\mathbb{Q} \to F$.