2012/10/03 by Wolfdieter Lang, Lang, Wolfdieter · 2 citations
Computer Science · Mathematics · #11R32 (Primary) 08B10 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1210.1018
openalex publication_date 2012/10/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.