TITLE: How to get the Galois group for $R/Q$
QUESTION [7 upvotes]: Let $R$ and $Q$ be the real number field and the rational number field, how to get the Galois group of $R/Q$?

REPLY [12 votes]: Let $f\in Aut_{\mathbb{Q}}(\mathbb{R})$.
Hints:
1) $\forall a,b\in\mathbb{R} [a\leq b \rightarrow f(a)\leq f(b) ]$. To show this, note that $f(b)-f(a)=f(b-a)=f((\sqrt{b-a})^2)=f(\sqrt{b-a})^2\geq 0$
2) Let $x$ be any real number. Let $r_n$ be an increasing sequence of rationals that converge to $x$. we find that $\forall n\in \mathbb{N} [r_n=f(r_n)\leq f(x)]$. By taking limits, we find that $x\leq f(x) $
3) Let $s_n$ be any decreasing sequence that converges to $x$. imitate the argument of step 2 to deduce that $x\geq f(x)$ for every real $x$
Thus, $f$ is the identity. Hence $Aut_{\mathbb{Q}}(\mathbb{R})$ is trivial.