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In the very beginning of baby Rudin we are given

$\textbf{1.19 Theorem}$ There exists an ordered field $\textit{R}$ with the least upper bound axiom. Moreover, $\textit{R}$ contains $\textit{Q}$ as a subfield.

Are the real numbers the $\textit{unique}$ ordered field with l.u.b. property with the rationals being a subfield? Of course we know $\mathbb{R}$ exists through many constructions, but are there any uncountably infinite fields with the same properties that aren't $\mathbb{R}$ up to isomorphism?

My intuition is telling me that it's true since the fact that $\mathbb{Q}$ (or an isomorphic copy) being a subfield of each must mean that both fields must capture the same properties. But I'm really not sure. Apologies if this question is silly.

Good Morning Captain
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    None of the fields that don't admit $\mathbb Q$ as a subfield can be ordered, so we only have to speak of characteristic 0 (i.e., fields that admit $\mathbb Q$ as a subfield) fields. Among these fields, it indeed is true that $\mathbb R$ is the unique field upto isomorphism admitting an order structure and satisfying the least upper bound property. – Aniruddh Agarwal Jun 02 '19 at 21:56
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    No this is a good question. This gives a bigger picture answer to your question. – Rushabh Mehta Jun 02 '19 at 21:59
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    It may also be of interest that the multiplicative structure is irrelevant. Any two complete and densely ordered abelian groups are isomorphic. The multiplicative structure of the reals can be recovered from the additive and order structures. – Rob Arthan Jun 02 '19 at 22:04
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    What is “baby Rudin”? – gen-ℤ ready to perish Jun 02 '19 at 22:40
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    @ChaseRyanTaylor: Rudin's Principles of Mathematical Analysis has acquired the irritating soubriquet "baby Rudin" in US mathematics culture. – Rob Arthan Jun 02 '19 at 23:03
  • @RobArthan : Interesting, +1. Any references? – MPW Jun 03 '19 at 00:02
  • @MPW:@book{ebbinghaus90, author="H.-D. Ebbinghaus and H. Hermes and F. Hirzebruch and M. Koecher and K. Mainzer and J. Neukirch and A. Prestel and R. Remmert", title="{Numbers}", publisher=SV, year=1990} is an accessible modern reference. – Rob Arthan Jun 03 '19 at 00:43
  • @RobArthan I didn't realize the terminology was irritating to people ... I guess I will not refer to it as baby Rudin anymore – Rushabh Mehta Jun 05 '19 at 19:07
  • @DonThousand: don't worry too much: "baby Rudin" is very low on my list of life's irritants! – Rob Arthan Jun 05 '19 at 19:41

2 Answers2

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Yes. The relevant property is not, however, that $\mathbb{Q}$ is a subfield, but rather the fact of the supremum property (or any one of any number of other statements equivalent to Dedekind completeness). $\mathbb{R}$ is the only ordered field (up to isomorphism) with this property.

The_Sympathizer
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Yes, the real numbers are unique in the sense you described. See the epilogue of Spivak's Calculus (chapter 29 or 30, depending on the edition).

Sambo
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