A055035 Degree of minimal polynomial of sin(Pi/n) over the rationals.

1, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 4, 12, 3, 8, 8, 16, 3, 18, 8, 12, 5, 22, 8, 20, 6, 18, 12, 28, 4, 30, 16, 20, 8, 24, 12, 36, 9, 24, 16, 40, 6, 42, 20, 24, 11, 46, 16, 42, 10, 32, 24, 52, 9, 40, 24, 36, 14, 58, 16, 60, 15, 36, 32, 48, 10, 66, 32, 44, 12, 70, 24, 72

Offset: 1

Author

Shawn Cokus (Cokus(AT)math.washington.edu)

Keywords

• nonn
• easy

Comments

Also degree of minimal polynomial of function F(n)=(gamma(1/n)*gamma((n-1)/n))/Pi over the rationals. Roots of minimal polynomials of F(n) belonging to algebraic extension of sin(n/Pi) and vice versa (e.g. gamma(1/11)*gamma(10/11)/Pi = 20*sin(Pi/11) - 112*sin(Pi/11)^3 + 256*sin(Pi/11)^5 - 256*sin(Pi/11)^7 + (1024*sin(Pi/11)^9)/11). - Artur Jasinski, Oct 17 2011

The algebraic numbers sin(Pi/(2*l)) are given in A228783 in the power basis of the number field Q(2*cos(Pi/(2*l))) if n is even and of Q(2*cos(Pi/l)) if l is odd. In A228785, sin(Pi/(2*l+1)) is given in the power basis of Q(2*cos(Pi/(2*(2*l+1)))) (only odd powers appear). The minimal polynomials for 2*sin(Pi/n), n>=1, are given in A228786. - Wolfdieter Lang, Oct 10 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..175
• Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
• Eric Weisstein's World of Mathematics, Trigonometry Angles

Crossrefs

Cf. A000010, A228786 (row length), A093819.

Programs

Mathematica

a[n_] := If[n==2, 1, EulerPhi[n]/{1, 1, 2, 1}[[Mod[n, 4]+1]]]; Table[a[n], {n, 80}]
a[n_] := Exponent[ MinimalPolynomial[Sin[Pi/n]][x], x]; Array[a, 75] (* Robert G. Wilson v, Jul 28 2014 *)

Formula

a(1)=1, a(2)=1, a(n)=phi(n)/(1, 1, 2, 1 for n=0, 1, 2, 3 mod 4) for n>2, where phi is Euler's totient, A000010

a(n) = A093819(2*n), n >= 1.- Wolfdieter Lang, Oct 29 2019