Everyone is aware of the standard result from undergraduate field theory that a real number $\alpha$ is constructible by straightedge and compass if and only if there exists a finite sequence of field extensions $\mathbb{Q} = K_0 \subset K_1 \subset \cdots \subset K_n$ such that $\alpha \in K_n$ and $[K_i : K_{i-1}] = 2$ for $1 \leq i \leq n$. I'm assuming that other equivalent conditions for the constructibility of reals have emerged since Pierre Wantzel's paper in the 1830s, but the textbooks that I have encountered thus far all seem to provide the standard proof involving the condition that $[\mathbb{Q}(\alpha) : \mathbb{Q}] = 2^n$ for some $n \geq 0$. My question is simple - what are some alternative equivalence results that are now available to us?
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1$\begingroup$ It is constructible iff it is contained in every subfield of the reals where all positive numbers have square roots — is that helpful? $\endgroup$– user44143Feb 9, 2023 at 20:18
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$\begingroup$ @MattF. Yes that is helpful, many thanks for your reply. I am basically interested in cataloguing every known equivalence result for the constructibility of real numbers - especially anything that is an not obvious corollary of the textbook algebraic formulation above. $\endgroup$– Menander IFeb 10, 2023 at 7:08
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