Converse of the theorem "If a real number $c$ is constructible, then $c$ is algebraic of degree a power of 2 over the field $\mathbb{Q}$ of rationals" In Thomas W. Hungerford's ALGEBRA book Chapter V, Proposition 1.16 (page # 239) states that

If a real number $c$ is constructible, then $c$ is algebraic of degree a power of 2 over the field $\mathbb{Q}$ of rationals.

Our Galois theory lecturer said that the converse of this theorem is not true. That is there is at least one algebraic number (say $c_{0}$) which is algebraic of degree a power of 2 over the field $\mathbb{Q}$ of rationals but it is not constructible. So I tried to find such an algebraic number. But I'm having trouble finding a counterexample. Can any one please give me a hint or an idea ?
Any hints/ideas are much appreciated. Thanks in advance for any replies.

You need to find a minimal polynomial of degree a power of 2, but with splitting field degree having other prime factors.
Here is an example (from here):
$x^4 + x + 1$
On MO there is another example.
