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

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

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

Positivity: if $x=p\mathrm{/}q>0$ 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: $\iota (1)=\iota (1\mathrm{/}1)=\iota (1)(1\cdot 1_F{)}^{-1}=1_F$; 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 $x\mathrm{\ne }y$ then $x<y$ or $y<x$, and 3.2 forces $\iota (x)\mathrm{\ne }\iota (y)$; so $\iota$ is injective, an embedding of ordered fields.

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