TITLE: What's the automorphism group of the real and complex numbers and the quaternions?
QUESTION [14 upvotes]: According to Wikipedia the automorphism group of the octonions is the exceptional group $G_2$. Are there analogous groups for the real numbers, the complex numbers and the quaternions?

REPLY [2 votes]: Regarding $R,$ let $h:F \to  G$ be an isomorphism between subfields $F,G$ of $R .$
If  $\forall x\in F\; (x>0\implies \sqrt x\in F),$ then $h=id_F. $ Because, for $x,y\in F,$ we have $$x>y\implies 0\ne h(x)-h(y)=(h(\sqrt {x-y}))^2 \implies h(x)>h(y).$$ And $h|Q=id_Q.$ So $\{q\in Q:q<x\}=\{q\in Q:q<h(x)\}$ for all $x\in F.$ In particular if $F=R$ then  $h=id_R$  and $G=R.$