Uniqueness of $\mathbb{R}$

Question

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.

Answer 1

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.

Answer 2

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).