(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find a field that is an ordered field in two distinct ways.

2. Relevant equations

The set F of all numbers of form r + s√2 , where r,s ε Q and the operations of addition and multiplication are those of the real number system ℝ of which F is a subset, is an ordered field in that there is a subset P of F such that addition and multiplication are closed in P, and x ε F →

exactly one of the following:

x ε P; x=0; -x ε P

This means that P is the set of all members of F that are positive members of ℝ, so P is the positive reals in F.

3. The attempt at a solution

A second way in which F can be ordered is by way of the subset B, such that

r + s√2 ε B iff r - s√2 ε P .

This I find troublesome, as (I am under the impression) that this implies that B = ∅.

My reasoning for this is that numbers in B will end in the digits that r - s√2 will end in (which would be, e.g., of form 1 - √2), and these are different than the numbers that r + s√2 span (for instance, 1 + √2).

Is B=∅? Are there faults in my reasoning?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A field that is an ordered field in two distinct ways

**Physics Forums | Science Articles, Homework Help, Discussion**