Cyclotomic Extensions and Constructibility

A primitive nth root of unity has multiplicative order exactly n. Its powers generate all nth roots, so adjoining one to Q gives the splitting field of x to the n minus one. The primitive roots are precisely the powers indexed by integers relatively prime to n. Their product polynomial is the cyclotomic polynomial Phi_n, which is irreducible over Q and has degree Euler phi of n.

An automorphism must send a chosen primitive root zeta to another primitive root, zeta to the a, where a is invertible modulo n. Every such assignment occurs, and composition multiplies the exponents modulo n. The Galois group is therefore the modular unit group, which is abelian but need not be cyclic. For n equal eight, every odd residue squares to one; the group has four elements yet no element of order four.

Starting from a unit segment and rational coordinates, intersections of lines require linear equations. A line with a circle, or two circles after subtracting their equations, leads to a quadratic equation. Consequently every constructible real coordinate lies in a tower whose successive degrees are at most two. Conversely, arithmetic operations and extraction of square roots of positive constructible quantities can be implemented geometrically, so membership in such a real quadratic tower characterizes constructible coordinates.

This gives a useful necessary test: the minimal polynomial degree of a constructible real algebraic number must be a power of two. For a general algebraic number, the degree condition alone is not sufficient; the relevant normal closure must permit the required quadratic tower. For roots of unity, the cyclotomic Galois structure supplies that extra control. A regular n-gon is constructible exactly when phi(n) is a power of two, for n at least three.

Writing n as a product of prime powers makes the condition concrete. Euler's formula multiplies p to the e minus one times p minus one for each prime-power factor. To leave only powers of two, every odd prime factor must occur once and must have p minus one a power of two. Such primes are Fermat primes. Therefore n must be a power of two times a product of distinct Fermat primes; the condition refers to primes, not every number of Fermat form.

The same reasoning explains familiar impossibilities. Duplicating a cube requires cube root of two, which has degree three by Eisenstein's criterion and cannot lie in a quadratic tower. An arbitrary angle cannot always be trisected because some trisection equations give irreducible cubics. These are statements about exact constructions with the specified ideal tools. They do not forbid numerical approximation, marked rulers, origami operations, or special angles whose trisectors happen to be constructible.