How to proof that a field is complete order field?

Question

I know what an ordered field is but how to actually proof that a field, for example $Q$ (rational numbers), is an ordered field?

Answer 1

You have to define an order on $\mathbb{Q}$ and show that it satisfies the required conditions. For this one goes back and recalls how you defined the rational numbers. As far is I know one usually defines it as the field of fractions of $\mathbb{Z}$, i.e. as $(\mathbb{Z} \times \mathbb{Z} )/\sim$ where $(a,b)\sim (c,d)$ iff $ad=bc$. You can define an order on $\mathbb{Q}$ as follows: We say $[(x,y)] \leq [(z,w)]$ if for any two representatives $(a,b)$ and $(c,d)$(of $[(x,y)]$ resp. $[(z,w)]$) s.t. $c,d>0$ we have $ad\leq bc$. Now you can easily verify that this is well defined and defines a total order on $\mathbb{Q}$.