Question

To prove: D^+ subseteq Q^+ Given information: text{Each} x in D text{is identified with} [x,e] text{in Q}

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asked 2020-11-27
To prove: \(\displaystyle{D}^{+}\subseteq{Q}^{+}\)
Given information:
\(\displaystyle\text{Each}\ {x}\ \in{D}\ \text{is identified with}\ {\left[{x},{e}\right]}\text{in Q}\)

Answers (1)

2020-11-28

Formula used:
1) An ordered field is an ordered integral domain that is also a field.
2) In the quotient field Q of an ordered integral domain D, defined \(\displaystyle{Q}^{+}\) by
\(Q^+=\left\{[a,b]\mid ab \in D^+\right\}\)
3) Well-ordered \(\displaystyle{D}^{{+}}:\)
If D is an ordered integral domain in which the set \(\displaystyle{D}^{{+}}\) of positive elements is well-ordered, then e is the least element of \(\displaystyle{D}^{{+}}.\)
Proof:
\(\displaystyle\text{Let}\ {x}\in{D}^{+},\text{then}\ {x}\in{D}\)
\(\displaystyle\text{So},{\left[{x},{e}\right]}\text{in Q where e is the least element of}\ {D}^{+}\)
\(\displaystyle\text{Now}\ {e}\in{D}^{+}\text{and}\ {x}\in{D}^{+}\text{implies that}\ {x}{e}\in{D}^{+}\)
\(\displaystyle\text{Therefore},\ {\left[{x},{e}\right]}\in{Q}^{+}\subseteq{Q}\)
\(\displaystyle\text{So},\ {x}\in{Q}^{+}\)
\(\displaystyle\text{Hence},{D}^{+}\subseteq{Q}^{+}\)

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