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    $\begingroup$ What is your definition of $\mathbb{C}$? $\endgroup$ Commented Apr 24, 2021 at 10:17
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    $\begingroup$ @Oniqa One can define $\mathbb{C}$ as an algebraic closure of $\mathbb{R}$. It is unique only up to isomorphism. Explicitly one can take $\mathbb{R}[\alpha_1,\alpha_2]/(\alpha_1+\alpha_2,\alpha_1 \alpha_2 -1)$ and then $\{\alpha_1,\alpha_2\} = \{\pm i\}$. $\endgroup$ Commented Apr 24, 2021 at 10:53
  • $\begingroup$ @FrançoisBrunault, what would it even mean to have a canonical isomorphism to a structure defined only up to isomorphism? $\endgroup$ Commented Apr 26, 2021 at 15:18
  • $\begingroup$ @LSpice Indeed, this quotient of $\mathbb{R}[\alpha_1,\alpha_2]$ is just one algebraic closure. There is no canonical isomorphism with the algebraic closure someone else may come up with, like $\mathbb{R}[x]/(x^2+1)$. $\endgroup$ Commented Apr 26, 2021 at 16:07