The standard way of proving that a given construction cannot be done using compass and straightedge alone starting from the complex numbers 0 and 1 is to show that, if it could be done, then it would lead us to a point _z_ such that there is no tower of fields
Q = K_0 <= K_1 <= K_2 <= ... <= K_n
with _z_ in K_n and such that the degree of each field over its predecessor is 1 or 2. This shows that if _z_ transcendent or if it is an algebraic number whose degree is no a power of 2, then _z_ is not constructible. The most famous cases of impossible constructions which lead to an algebraic number are the impossibility of trisecting a 60 degree angle and of doubling a cube; in each case, the standard proof leads to an algebraic number of degree 3.
Can anyone tell me where to find examples of impossible constructions which lead to algebraic numbers of degree four (or, more generally, some power of 2)?