What is the Definition of an Ordered Field?

[Buri]

Hi I've looked up the definition of an ordered field and I've found two slightly different ones. One is that an ordered field is a field with a total order, but I've found another which requires it to also satisfy:

(1) If a <= b then a + c <= b + c
(2) If 0 <= a and 0 <= b, then 0 <= ab.

Which one is the correct one? And could someone give me an example of a field which has a total ordering but doesn't satisfy the extra conditions? Thanks I'd appreciate it :)

 

[HallsofIvy]

It would make no sense to define an "ordered field" as simply a field with an order. Any time you put together two concepts, such as "order" and "field", you must have some rules for how they work together, which is what your (1) and (2) do.

 

[WWGD]

I think the complex numbers with the dictionary ordering are an example, but I'm not sure.