constructible

Constructible is a mathematical term used to describe objects that can be created using a straightedge and compass from a given set of starting objects. In Euclidean geometry, a length, point, or figure is called constructible if it can be obtained from the given units by a finite sequence of classical constructions.

A central notion is a constructible number: a real number that can be constructed from the rational numbers using a finite sequence of field operations (addition, subtraction, multiplication, division) and square root extractions. Equivalently, constructible numbers lie in a field obtained from the rationals by successive quadratic extensions. Consequently, every constructible number is algebraic and its minimal polynomial over Q has degree a power of two. Rational numbers, square roots, and many combinations thereof are constructible, while general cubic roots are not, reflecting the historical limits of ruler-and-compass constructions.

A classic consequence is the theory of constructible polygons. A regular n-gon is constructible with straightedge and compass if and only if n is of the form 2^k times a product of distinct Fermat primes (the known Fermat primes being 3, 5, 17, 257, and 65537). Therefore, polygons with 3, 4, 5, 6, 8, 9, 10, 12, 15, 17-gons, among others, are constructible, while many other n-gons are not.

In a broader sense, constructible also appears in algebraic geometry, where a constructible set is a subset of an algebraic variety that can be written as a finite union of locally closed sets in the Zariski topology.