1. ## Cyclotomic field

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.

2. CAn somebody help me with this?

3. Originally Posted by ZetaX

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.
well, $\displaystyle K=\bigcup (\mathbb{Q}_{2^n} \cap \mathbb{R})$ is normal because $\displaystyle \mathbb{Q}_{2^n}/\mathbb{Q}$ is abelian and thus each $\displaystyle K_n=\mathbb{Q}_{2^n} \cap \mathbb{R}$ is normal. i'm not expert in infinite Galois theory but i'm pretty sure the key to the second part of your

problem is this isomorphism: $\displaystyle \text{Gal}(K/\mathbb{Q}) \cong \varprojlim \text{Gal}(K_n/\mathbb{Q}).$ now the question is what exactly is $\displaystyle \text{Gal}(K_n/\mathbb{Q})$ ?