Let $\displaystyle n>2$ and let $\displaystyle \mathbb Q_{2^n}$ be the

$\displaystyle 2^n-$th cyclotomic field.

Show that $\displaystyle \mathbb Q_{2^n} \cap \mathbb R$ is a normal extension and its Galois group is cyclic of order $\displaystyle 2^{n-2}.$