- #1

FallMonkey

- 11

- 0

## Homework Statement

Is the ordering of a ordered field unique? That is, is it possible to have different ordering set(the order), we call P1 and P2, both able to make a field F into a ordered field?

## Homework Equations

no.

## The Attempt at a Solution

First I tried to assume now there's a X in P1, but not in P2(because P1 != P2), and therefore by definition -X must be in P2. After that, I tried to get to a point where I can make X also in P2 to have contradiction. But I can't.

Then I tried to go from ration numbers. With some existing theorem, Q(rational number set) is the smallest ordered field, and a subfield of any ordered field. Therefore if there're such P1 and P2 in some ordered field, they will both have Q, thus the element that is not shared can not be a rational number. After that, I stopped without any idea how i could progress.

Thank you very much for viewing, and let me know anything I unclearly stated.