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 Post subject: CompletenessPosted: Tue Dec 29, 2015 11:31 pm
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Does the ordered field of the rational functions satisfy the completeness theorem : " All non-empty

sets have a supremum" .

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 Post subject: Re: CompletenessPosted: Sat Jan 30, 2016 6:19 am

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
Papapetros Vaggelis wrote:
Does the ordered field of the rational functions satisfy the completeness theorem : " All non-empty

sets have a supremum" .

I give the answer but not the solution so that I don't spoil someone's fun if he/she wants to try.

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Imagination is much more important than knowledge.

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 Post subject: Re: CompletenessPosted: Wed May 18, 2016 7:51 pm

Joined: Sat Nov 14, 2015 6:32 am
Posts: 159
Location: Melbourne, Australia
Papapetros Vaggelis wrote:
Does the ordered field of the rational functions satisfy the completeness theorem : " All non-empty

sets have a supremum" .

No, it does not, because the ordered field of the rational functions does not satisfy the Archimedean property. This is quite known.

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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