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It is well known that there are wild automorphisms of the complex numbers, and that defining the topology on the complex numbers gets rid of all these wild automorphisms. Using this fact, I proved that any nontrivial automorphism of the field of complex numbers does not preserve the set of complex numbers with positive imaginary part, a set closed under addition (By "preserve" a subset I mean that the image of the subset under the automorphism should be the subset itself). I also proved that any nontrivial automorphism of the field of complex numbers does not preserve the set of complex numbers that is a semicircular arc with radius $0.5$ combined with a filled circle of radius $0.25$ centered at the origin, a set closed under multiplication.

My question: is there any such automorphism-killing subset of the complex numbers that is closed under both addition and multiplication?

mathlander
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  • Well, $\mathbb{C}$ is such a subset, so probably you need an assumption excluding that case. – radekzak Jan 30 '23 at 19:27
  • @radekzak I do not mean "preserves each element of the subset." I mean that each element of the subset should map to an element of the subset, and each element outside the subset should map to an element outside the subset. I edited my question to clarify what I mean. – mathlander Jan 30 '23 at 19:29

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