Show that the union of $\displaystyle \mathbb Q_{2^n}\cap \mathbb R$, where $\displaystyle n$ is from $\displaystyle 3$ to $\displaystyle \infty $, is a (profinite) normal extension of $\displaystyle \mathbb Q.$

And show that its Galois group is the additive group of the integer 2-adic number.