Construction, A Tower of Quadratic Extensions

A Tower of Quadratic Extensions

In the last section we showed that the set of constructible points forms a field within the reals. You can think of these real numbers as coordinates in the plain, or as established distances between designated points. Either is equivalent to the other. If a distance r is established, move that distance from an axis to set a coordinate equal to r. Conversely, if a point has a coordinate of r, measure the perpendicular distance to the axis to find a distance r.

Given a distance r, we know how to take the square root of r. Therefore all points in the field Q(sqrt(r)) are constructible, for any positive value of r. Then take any real number s from this field and adjoin its square root. This creates another field extension of dimension 2. You can continue this process indefinitely, building a tower of quadratic extensions.

Let's look again, with a little more rigor. Let z be a real number that is contained in a field extension of dimension 2^m. This extension contains an intermediate field of dimension 2^m-1, which contains a field extension of dimension 2^n-2, all the way down to the rationals. Now z is the root of an irreducible quadratic over the previous field extension of dimension 2^m-1. By induction, everything in this penultimate extension is constructible. Apply the quadratic formula, and z becomes accessible, once we take the square root of the discriminant. Square roots can be constructed, so z can be constructed. Any point that is in a tower of field extensions, each having dimension 2 over the previous, is constructible.

Now let's look at the converse. Each construction operation locates another point in the established field of points, or it extends the field. If the extension is always quadratic, we're done.

Consider the available techniques for identifying an additional point. A new point might be the intersection of two lines derived from two pairs of previously designated points. Find the intersection by solving two simultaneous linear equations in the plane. Use cramer's rules if you like. This incorporates only field operations, so the new point lies in the preexisting field. This does not create a field extension at all.

Another possibility is the intersection of a circle and a line. Write the equation of the circle as:

(x-x₀)² + (y-y₀)² = r²

Replace y with a linear function of x. The result is a quadratic equation in x. This can only create a quadratic field extension. The new point x, identified by construction, generates a field extension of dimension two.

The last possibility is the intersection of two circles. Write the two equations and subtract one from the other. The squared terms cancel, leaving a linear equation. This is, in fact, the equation of the line that passes through the two points of intersection. Intersect this line with either of the two circles, using the procedure described above. Once again the new points generate a field extension of dimension 2.

In summary, z is constructible iff it belongs to a tower of quadratic field extensions.