Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$

Question

Is there any subfield of $\mathbb{C}$ which is isomorphic to $\mathbb{R}$ but not $\mathbb{R}$ ?

Answer 1

Let $B$ be a trascendence basis of $\mathbb R$ over $\mathbb Q$, so that $\mathbb R$ is an algebraic extension of $\mathbb Q(B)$. The set $B$ is infinite: let $B'$ be the result of removing one element. Since $B$ and $B'$ are in bijection, there is an algebraic extension of $\mathbb Q(B')$ contained in $\mathbb C$ which is isomorphic to $\mathbb R$, and it is not $\mathbb R$ because it does not contain the element we removed.