Straightedge and compass construction

Summary

A straightedge and compass construction draws points, lines and circles in the plane from given points with two ideal tools. The straightedge draws the line through two constructed points, and it has no marks. The compass draws the circle with a constructed centre through a constructed point. New points are the intersections of the lines and circles drawn so far. The constructible numbers describe the points that these steps can reach.

Remarks

Each step solves linear or quadratic equations. So the coordinates of the new points lie in an extension of degree $1$ or $2$ of the field of the old coordinates. This turns questions about constructions into questions about degrees of field extensions. It shows that doubling the cube, trisecting the angle and squaring the circle are impossible with these tools.

Explanations

L1 human

The ancient Greeks asked which figures can be drawn with an unmarked ruler and a compass alone. Field theory gives the answer: a length can be drawn exactly when it comes from rational numbers by field operations and square roots.