I'm trying to prove that the field $Q$ (the rationals) is ordered using the order axioms for a field.
The order axioms for a field $F$ with $a,b,c \in F$:
- For $a$ and $b$ only one of the below can be true:
i) $a<b$
ii) $b<a$
iii) $b=a$
If $a<b$ and $b<c$ then $a<c$.
If $a<b$ then $a+c<b+c$.
If $a<b$ then $ac<bc$ for 0
I started by defining 1.i),ii),iii) for $\frac{a}{b}, \frac{c}{d} \in Q$. Then I tried to prove that $\frac{a}{b}, \frac{c}{d}, \frac{e}{f} \in Q$ satisfy properties 2.,3, and 4. but the proofs for all of them didn't really work out since 2.,3. and 4. all seem too axiomatic. For instance for 2., I started with $\frac{a}{b}<\frac{c}{d}$ and $\frac{c}{d}<\frac{e}{f}$ and said $\frac{a}{b}<\frac{c}{d}<\frac{e}{f} \to \frac{a}{b}<\frac{e}{f}$ but I'm not sure if this is a proof more than it is a simple restatement of the axiom. My other proofs were very similar so I'm wondering if there's any special tricks or anything which I have to do.