What is the degree of a real closure of an ordered field?

field-theory extension-field model-theory cardinals ordered-fields

Proposition 5 in https://arxiv.org/abs/2007.13550 (and the preceding paragraph) answers Question I. There exists a proper subfield $F$ over which $\mathbb{R}$ is algebraic, such that $[\mathbb{R} : F] = \aleph_0$.

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