Suppose we work in a model in which the Axiom of Choice does not hold, and in which $\mathbb{C}$ only has one nontrivial automorphism (such models exist).

**Question**: *"how many" subfields of $\mathbb{C}$ are isomorphic to $\mathbb{R}$?*

More precisely: in what way does the size of the automorphism group of $\mathbb{C}$ influence/control this question?

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