Page "Compass and straightedge constructions" Paragraph 38

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of addition, subtraction, multiplication, division, complex conjugate, and square root, which is easily seen to be a countable dense subset of the plane.

Each of these six operations corresponding to a simple compass and straightedge construction.

From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations.

More efficient constructions of a particular set of points correspond to shortcuts in such calculations.