constructible number

The ancient Greeks built geometry with only two tools: an unmarked straightedge and a compass. Starting from two points a unit apart, you can draw lines and circles, mark their intersections, and so create new points. A constructible number is any length you can reach this way. The surprise is that this purely geometric game has a precise algebraic description, and that description settles famous problems the Greeks could not.

Precisely, a real number is constructible if it can be obtained from the rationals by a finite sequence of additions, subtractions, multiplications, divisions, and square roots — because intersecting lines and circles solves only linear and quadratic equations. Algebraically, a is constructible if and only if it lies in a field at the top of a tower Q = K_0 subset K_1 subset ... subset K_n with each [K_{i+1} : K_i] = 2. In particular every constructible number is algebraic of degree a power of 2 over Q.

This degree obstruction instantly resolves three classical impossibilities. Doubling the cube needs 2^(1/3), of degree 3, not a power of 2. Trisecting a general angle needs a root of an irreducible cubic. Squaring the circle needs pi, which is transcendental, hence not algebraic at all. Regular n-gon constructibility (the Gauss-Wantzel theorem) reduces to whether the relevant cyclotomic degree is a power of 2.