constructibility

Constructibility is a concept in geometry and algebra concerning whether objects, lengths, or numbers can be produced using only a straightedge and compass from a given unit length. The classical focus is on which geometric figures and lengths are attainable through a finite sequence of allowable operations.

In the standard Euclidean setting, a length is constructible if it can be obtained from the unit length by a finite sequence of additions, subtractions, multiplications, divisions, and square root extractions. In algebraic terms, a complex number is constructible if its real and imaginary parts lie in a field obtained from the rationals by a tower of quadratic (i.e., degree-2) extensions. Equivalently, the minimal polynomial of a constructible number over Q has degree a power of two.

One key consequence of this theory is the characterization of constructible regular polygons. A regular n-gon is constructible with straightedge and compass if and only if n = 2^k times a product of distinct Fermat primes (primes of the form 2^{2^m} + 1). The known Fermat primes are 3, 5, 17, 257, and 65537. Classical impossibilities such as trisecting arbitrary angles, constructing a cube with a given edge, and squaring the circle are explained by these degree constraints and show the limits of constructibility with the classical tools.

Beyond polygons, constructibility also informs the study of which numbers and figures can arise from sequences of quadratic extensions, providing a unifying framework for geometric and algebraic constructibility.