(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?

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# Homework Help: A field that is an ordered field in two distinct ways

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