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The following is a question I've been pondering for a while. I was reminded of it by a recent dicussion on the question How to tell $i$ from $-i$?

Can you find a field that is abstractly isomorphic to $\mathbb{C}$, but that does not have a canonical choice of square root of $-1$?

I mean canonical in the following sense: if you were to hand your field to one thousand mathematicians with the instructions "Pick out the most obvious square root of -1 in this field. Your goal is to make the same choice as most other mathematicians," there should be be a fairly even division of answers.

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    Am I missing something here? What is the canonical choice for $\sqrt{-1}$ in $\mathbb{C}$?2012-08-12
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    The definition of $\mathbb{C}$ I had in mind was the set of formal expressions $a+ib$, with $a,b\in\mathbb{R}$. The element called $i$ is then the canonical choice. Did you have a different definition of $\mathbb{C}$ in mind?2012-08-12
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    @Pink: the algebraic closure of $\mathbb{R}$?2012-08-12
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    $\mathbb{R}$ has many (isomorphic) algebraic closures. I was thinking of $\mathbb{C}$ as a particular choice of algebraic closure, given by a concrete description.2012-08-12
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    @Pink: then (as I suggested in my comment to Pete Clark's answer) you aren't talking about fields but about descriptions of fields.2012-08-12
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    All the answers so far work with a somewhat informal definition of “canonical” — hence their somewhat unsatisfactory debatability. One possible formalisation of the question is “Is there a definition in ZFC of a field, ZFC-provably isomorphic to C, but with no ZFC-definable square root of –1?” (Feel free to replace ZFC with your preferred foundational system, of course.)2012-08-12

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