Characterization of a subfield $K \varsubsetneq \mathbb {C}$ and $x\in \mathbb{R}$ > Characterize $x \in \mathbb R$ such that there exist a subfield $K \varsubsetneq \mathbb C$ such that $K(x) = \mathbb C$. -All subfields $K$ of $\mathbb{C}$ contain $\mathbb Q$, then all $x\notin \mathbb Q$ -If there exist $K$, it is uncountable. In fact if $K(x)= \mathbb {C}$ is countable, which is absurd. But this is not enough to characterize all the x.., Thank you in advance,

Denote by $E$ the set of $x \in \mathbb{R}$ satisfying your conditions.

* If $x \in \mathbb{R}$ is transcendental over $\mathbb{Q}$ or algebraic not totally real, then $x \in E$.



> Take $y=i \pi$ if $x$ is transcendental or $y \
otin \mathbb{R}$ conjugate to $x$ if $x$ is algebraic not totally real. Using Zorn's lemma you can find an automorphism $\varphi$ of $\mathbb{C}$ such that $\phi(y)=x$. Take $K:=\phi(\mathbb{R})$. Then $K[x] = \varphi(\mathbb{R}[y])=\mathbb{C}$

* If $x$ is algebraic totally real, then $x \
otin E$.



> Unfortunately I don't know how to do it without Artin-Schreier. Recall that Artin-Schreier states that if $K \subset \mathbb{C}$ is such that $[\mathbb{C}:K]<\infty$ then there exists an automorphism $\varphi$ of $\mathbb{C}$ inducing an isomorphism $\varphi : \mathbb{R} \xrightarrow{\sim} K$.