$c$ be algebraic , constructible over $\mathbb Q$ , $N$ be the normal closure of $\mathbb Q(c)/\mathbb Q$ ; then $[N:Q]$ is a power of $2$?

Question

$c$ be algebraic over $\mathbb Q$ , $N$ be the normal closure of $\mathbb Q(c)/\mathbb Q$ ; if $c$ is constructible over rationals then how to show that $[N:\mathbb Q]$ is a power of $2$ ?

Answer 1

The field $N$ is $\Bbb Q(c_1,\ldots,c_n)$ where the $c_j$ are the conjugates of $c$. Now $c$ lies in a tower of quadratic extensions of $\Bbb Q$. Therefore all the $c_j$ are also. Putting all these towers together we get that $N$ is an iterated quadratic extension of $\Bbb Q$.