The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon

TL;DR

This paper explores the algebraic and Galois-theoretic properties of the field generated by 2cos(pi/n), linking it to regular n-gons, Chebyshev polynomials, and introducing a new modular multiplication to analyze the Galois group structure.

Contribution

It provides explicit minimal polynomials, zeros, and Galois group analysis for the field Q(2cos(pi/n)), introducing a novel modular multiplication method called Modd n.

Findings

Derived product formulas for length ratios in regular n-gons.

Presented explicit minimal polynomials for 2cos(pi/n).

Introduced the Modd n operation to analyze Galois groups.

Abstract

The normal field extension Q(rho(n)), with the algebraic number rho(n) = 2 cos(pi/n) for natural n, is related to ratios of the lengths between diagonals and the side of a regular n-gon. This has been considered in a paper by P. Steinbach. These ratios are given by Chebyshev S-polynomials. The product formula for these ratios was found by Steinbach, and is re-derived here from a known formula for the product of Chebyshev S-polynomials. It is shown that it follows also from the S-polynomial recurrence and certain rules following from the trigonometric nature of the argument x = rho(n). The minimal integer polynomial C(n,x) for rho(n) is presented, and its simple zeros are expressed in the power-basis of Q(rho(n)). Also the positive zeros of the Chebyshev polynomial S(k-1,rho(n)) are rewritten in this basis. The number of positive and negative zeros of C(n,x) is determined. The coefficient C(n,0) is computed for special classes of n values. Theorems on C(n,x) in terms of monic integer Chebyshev polynomials of the first kind (called here t-hat) are given. These polynomials can be factorized in terms of the minimal C-polynomials. A conjecture on the discriminant of these polynomials is made. In order to determine the cycle structure of the (Abelian) Galois group a novel modular multiplication, called Modd n is introduced. On the reduced odd residue system Modd n this furnishes a group which is isomorphic to this Galois group.