Find a non-constructible algebraic number of degree $4$ over $\Bbb Q$

Question

I cannot think of a non-constructible algebraic number of degree $4$ over $\Bbb Q$ so far. I wish if I can find such an example. Could some one tell me some such numbers with justification? Also I would like to know the track of working out such an example. Any help or reference would be appreciate. Thanks in advance!

Answer 1

One problem which leads to such a number is Alhazen's billiard problem.

Answer 2

Take an $S_4$ extension which is the splitting field of a quartic polynmial, say $f(x)=x^4-4x+2$ with splitting field $K$. If the roots of $f(x)$ were constructible, then all the elements of $K$ would be constructible. For $G$ a Sylow $2$-subgroup of $S_4$, the fixed field of $G$, $K^{G}$ has odd degree over $\Bbb{Q}$, so the elements of $K\setminus\Bbb{Q}$ can't be constructible. From Milne, Remark 3.26, Fields and Galois Theory.