Examples of elementary substructures of $(\mathbb{C},0,1,+,-, \times)$

Question

I wonder what are some examples of elementary substructures of $(\mathbb{C},0,1,+,-, \times)$. I know that $(\mathbb{R},0,1,+,-, \times)$, $(\mathbb{Q},0,1,+,-, \times)$ are not since they don't satisfy $x^2=-1$.

Thanks!

Answer 1

As commenters have mentioned, any elementary substructure of $\mathbb{C}$ must satisfy the same first order properties. But it is well known that $\mathsf{ACF}_0$ (the theory of Algebraically Closed Fields of characteristic $0$) is model complete. This follows, for instance, from quantifier elimination.

How is this helpful?

Any elementary substructure of $\mathbb{C}$ must, in particular, model $\mathsf{ACF}_0$, since it is clear that $\text{Th}(\mathbb{C}) \supseteq \mathsf{ACF}_0$. But by model completeness, we get the converse: any submodel $\mathfrak{M}$ of $\mathbb{C}$ that models $\mathsf{ACF}_0$ must be elementary!

So we're looking for the subrings of $\mathbb{C}$ which are themselves algebraically closed fields of characteristic $0$.

We see one obvious elementary substructure: $\overline{\mathbb{Q}}$. However, there are lots more! For a more detailed discussion about the algebraically closed subfields of $\mathbb{C}$, see here for instance.

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I hope this helps ^_^