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The normal field extension Q(rho(n)), with the algebraic number rho(n) = 2cos(pi/n) for natural n, is related to ratios of the lengths between diagonalsand 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 productformula for these ratios was found by Steinbach, and is re-derived here from aknown formula for the product of Chebyshev S-polynomials. It is shown that itfollows also from the S-polynomial recurrence and certain rules following fromthe trigonometric nature of the argument x = rho(n). The minimal integerpolynomial C(n,x) for rho(n) is presented, and its simple zeros are expressedin the power-basis of Q(rho(n)). Also the positive zeros of the Chebyshevpolynomial S(k-1,rho(n)) are rewritten in this basis. The number of positiveand negative zeros of C(n,x) is determined. The coefficient C(n,0) is computedfor special classes of n values. Theorems on C(n,x) in terms of monic integerChebyshev polynomials of the first kind (called here t-hat) are given. Thesepolynomials can be factorized in terms of the minimal C-polynomials. Aconjecture on the discriminant of these polynomials is made. In order todetermine the cycle structure of the (Abelian) Galois group a novel modularmultiplication, called Modd n is introduced. On the reduced odd residue systemModd n this furnishes a group which is isomorphic to this Galois group.

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