Constructible number

A constructible number is the coordinate of a point obtainable from rational starting data by straightedge-and-compass constructions.

A point of the Euclidean plane is constructible if it can be obtained from $(0,0)$ and $(1,0)$ by finitely many straightedge-and-compass operations: draw a line through two obtained distinct points, draw a circle with obtained centre and radius equal to an obtained segment length, and retain isolated intersection points. Its two coordinates are constructible numbers. Conversely, constructible real coordinates can be placed on perpendicular axes and their coordinate lines intersected, giving the point. Coordinates created at each isolated intersection require only solving linear or quadratic equations over the current coordinate field; hence they lie in a real field extension obtained by a finite tower of quadratic extensions. Coincident curves supply no isolated new intersection points.

A constructible circle has a constructible point as its centre and radius equal to the distance between two constructible points. When those coordinates lie in a real field $F$, its equation $(x-a{)}^2+(y-b{)}^2=r^2$ has coefficients in $F$ because the squared radius is a sum of squared coordinate differences. Intersecting it with a constructible line reduces to a quadratic equation. Intersecting two distinct circles reduces to the same situation by subtracting their normalized equations to obtain the radical axis. Thus each isolated real intersection requires at most a real quadratic field extension.

A constructible line is a line through two distinct constructible points. If the available coordinates lie in a real field $F$, a line through $(a,b)$ and $(c,d)$ has equation $(d-b)(x-a)-(c-a)(y-b)=0$, with coefficients in $F$ and a nonzero normal vector. Intersecting two distinct such lines introduces no new field elements, because solving a nonsingular pair of linear equations uses only field operations.

A finite real extension $K/\mathbb{Q}$ is constructible when it is contained in a towerEquivalently, every element of $K$ can be obtained from rational numbers by field operations and successive square roots.

The real cyclotomic fieldis Galois of degree eight with cyclic Galois group $(\mathbb{Z}/17\mathbb{Z}{)}^{\times }/\{\pm 1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos (2\pi /17)$ is constructible.