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Algebraic degree of sin(2*Pi/n).
+10
14
1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, 12, 6, 8, 4, 16, 6, 18, 2, 12, 10, 22, 4, 20, 12, 18, 3, 28, 8, 30, 8, 20, 16, 24, 3, 36, 18, 24, 8, 40, 12, 42, 5, 24, 22, 46, 8, 42, 20, 32, 6, 52, 18, 40, 12, 36, 28, 58, 4, 60, 30, 36, 16, 48, 20, 66, 8, 44, 24, 70, 12, 72, 36, 40, 9, 60, 24
COMMENTS
The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011 This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011 Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011 See A181872/ A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013
REFERENCES
I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
FORMULA
a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)= A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - Wolfdieter Lang, Jan 09 2011 a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - Wolfdieter Lang, Nov 30 2013
MATHEMATICA
a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)
Coefficient array for integer polynomial version of minimal polynomials of sin(2*Pi/n). Rising powers of x
+10
3
0, 2, 0, 2, -3, 0, 4, -2, 2, 5, 0, -20, 0, 16, -3, 0, 4, -7, 0, 56, 0, -112, 0, 64, -2, 0, 4, -3, 0, 36, 0, -96, 0, 64, 5, 0, -20, 0, 16, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024, -1, 2, 13, 0, -364, 0, 2912, 0, -9984, 0, 16640, 0, -13312, 0, 4096, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 224, 0, -448, 0, 256, 2, 0, -16, 0, 16, 17, 0, -816, 0, 11424, 0, -71808, 0, 239360, 0, -452608, 0, 487424, 0, -278528, 0, 65536, -3, 0, 36, 0, -96, 0, 64
COMMENTS
The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...]. pi(n,x) := Sum_{m=0..d(n)} a(n,m)*x^m, n >= 1, is related to the (monic) minimal polynomial of sin(2*Pi/n), called Pi(n,x), by pi(n,x) = (2^d(n))*Pi(n,x), with the degree sequence d(n)= A093819(n), and Pi(n,x) is given in A181872/ A181873. Pi(n,x)=Psi(c(n),x) with the minimal polynomials Psi(n,x) of cos(2*Pi/n), and c(n):= A178182(n). The minimal polynomials of sin(2*Pi/n) are, e.g., treated in the Lehmer and Niven references. (Note the mistake in the Lehmer references explained in the W. Lang link.) The fundamental polynomials Psi(n,x) are also studied in the Watkins-Zeitlin reference, where a recurrence is given.
REFERENCES
I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons..
FORMULA
a(n,m) = [x^m]pi(n,x), n >= 1, m=0.. A093819(n), and pi(n,x) defined above in the comments.
EXAMPLE
[0, 2], [0, 2], [-3, 0, 4], [-2, 2], [5, 0, -20, 0, 16], [-3, 0, 4], [-7, 0, 56, 0, -112, 0, 64], [-2, 0, 4], [-3, 0, 36, 0, -96, 0, 64], [5, 0, -20, 0, 16], ...
pi(2,x) = (2^1)*Pi(2,x) = 2*Psi(c(2),x) = 2*Psi(4,x) = 2*x.
MATHEMATICA
ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Sin[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc/Last[cc])); Flatten[ Table[ ro[n], {n, 1, 18}]] (* Jean-François Alcover, Sep 28 2011 *)
Coefficients of the algebraic number 2*sin(2*Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.
+10
2
0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, -3, 0, 1, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, -3, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 0, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 5, 0, -5, 0, 1, 0, -7, 0, 22, 0, -13, 0, 2, 0, -3, 0, 1, 0, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 0, 0, 0, -4, 0, 5, 0, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1, 0, 0, -1, 1
COMMENTS
The relevant trigonometric identity (used in the D. H. Lehmer and I. Niven references, given in A181871) is 2*sin(2*Pi/n) = 2*cos(2*Pi*(1/n -1/4)) = 2*cos(Pi*abs(n-4)/(2*n)) = 2*cos(Pi*p(n)/q(n)), with gcd(p(n), q(n)) = 1 (fraction p(n)/q(n) in lowest terms). One finds p(n) = A106609(n-4), n >=4, with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n), n >= 1. See the comments on these two A-numbers. Therefore, 2*sin(2*Pi/n) = R(p(n), rho(q(n))), with rho(k) = 2*cos(Pi/k), and the R-polynomials (monic version of Chebyshev's T-polynomials) are given in A127672. It may happen that p(n), the degree of R, is >= delta(q(n)), the degree of the algebraic number rho(q(n)). Here delta(k) = A055034(k) is the degree of the minimal polynomial C(k, x) of rho(k) found under A187360. In this case one can reduce all rho(q(n)) powers >= delta(q(n)) with the help of the equation C(q(n), rho(q(n))) = 0. Thus the final result is 2*sin(2*Pi/n) = R(p(n), x) (mod C(q(n), x)) with x = rho(q(n)). Because R is an integer polynomial this shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(q(n))) of degree delta(q(n)). The power basis of Q(rho(q(n))) is <1, rho(q(n)), ..., rho(q(n))^(delta(q(n))-1)>. Therefore the length of row n of this table is delta(q(n)).
The values n for which mod C(q(n), x) is in operation for the given formula for 2*sin(2*Pi/n) are those for which delta(q(n)) - p(n) <= 0, that is n = 1, 2, 12, 15, 18, 20, 21, 24, 25, 27, 28, 30,...
For the minimal polynomials of 2*sin(2*Pi/n) see the coefficient table A231188.
FORMULA
a(n,m) = [x^m] (R(p(n), x) (mod C(q(n), x)), n >= 1, m = 0, 1, ..., delta(q(n)) - 1, where the R and C polynomials are found in A187360 and A127672, respectively. p(n) = A106609(n-4), n >=4, with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n). Powers of x = rho(q(n)) = 2*cos(Pi/q(n)) appear in the table in increasing order.
EXAMPLE
[0], [0], [0, 1], [2], [0, 1, 0, 0], [0, 1], [0, -3, 0, 1, 0, 0], [0, 1], [0, 5, 0, -5, 0, 1], ...
The table a(n,m) begins (the trailing zeros are needed to have the correct degree for Q(rho(q(n)))):
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1: 0
2: 0
3: 0 1
4: 2
5: 0 1 0 0
6: 0 1
7: 0 -3 0 1 0 0
8: 0 1
9: 0 5 0 -5 0 1
10: 0 -3 0 1
11: 0 -7 0 14 0 -7 0 1 0 0
12: 1
13: 0 9 0 -30 0 27 0 -9 0 1 0 0
14: 0 5 0 -5 0 1
15: 0 -7 0 22 0 -13 0 2
16: 0 -3 0 1
17: 0 13 0 -91 0 182 0 -156 0 65 0 -13 0 1 0 0
18: 0 -4 0 5 0 -1
19: 0 -15 0 140 0 -378 0 450 0 -275 0 90 0 -15 0 1 0 0
20: -1 1
...
--------------------------------------------------------------------------
n=1: 2*sin(2*Pi/1) = 0. rho(q(1)) = rho(2) = 2*cos(Pi/2) = 0 and p(1) = 3. R(3, x) = -3*x + x^3 and C(2, x) = x. Therefore R(3, x) (mod C(2, x)) = 0. The degree of C(2, x) is delta(2) = A055034(2) = 1. Here one should use 1 for the undefined rho(q(1))^0 in order to obtain a(1, 0) = 0. n=2: 2*sin(2*Pi/2) = 0; rho(q(2)) = rho(2) = 0; p(2) = 1, R(1, x) = x , C(2, x) = x and delta(2) = 1. Therefore R(1, x) (mod C(1, x)) = 0. Again, rho(2)^0 is put to 1 here, and a(2, 0) = 0.
n=5: 2*sin(2*Pi/5) = R(1, rho(10)) (mod C(10, rho(10)) =1* rho(10) (the degree of C(10,x) is delta(10) = 4, therefore the mod prescription is not needed). Therefore, a(5, 0) =0, a(5,1) =1, a(n, m) = 0 for m=2, 3.
n =11: 2*sin(2*Pi/11) = R(7, x) (mod(C(22, x)) with x = rho(22), because p(11) = 7 and q(11) = 22. The degree of C(22, x) is delta(22) = 10, therefore the mod restriction is not needed and R(7, x) = -7*x + 14*x^3 - 7*x^5 + x^7. The coefficients produce the row [0, -7, 0, 14, 0, -7, 0, 1, 0, 0] with the two trailing zeros needed to obtain the correct row length, namely delta(q(11)) = 10.
Coefficient table for the minimal polynomials of 2*sin(4*Pi/n). Rising powers of x.
+10
1
0, 1, 0, 1, -3, 0, 1, 0, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 14, 0, -7, 0, 1, -2, 1, -3, 0, 9, 0, -6, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, -3, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, -7, 0, 14, 0, -7, 0, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, -2, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1
COMMENTS
The length of row n is A232626(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 9, 3, 17, 7, 19, 5,... In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2.
For the power basis representation of 2*sin(4*Pi/n) in the algebraic number field Q(rho(q(2,n))), with q(2,n)) = A232625(n) and rho(m) := 2*cos(Pi/m), see A232629. Call the row polynomials of A232629 PB2(n,x) (power basis polynomial for the case k=2 in 2*sin(2*Pi*k/n)). The minimal polynomial of 2*sin(4*Pi/n), call it MP2(n, x), is obtained from the conjugates rho(q(2,n),j), j= 1, ... , delta(q(2,n)) = A232626(n), which are the zeros of C(q(2,n), x), the minimal polynomial of rho(q(2,n)) = rho(q(2,n),1) (for C see A187360). MP2(n, x) = product(x - PB2(n, rho(q(2,n),j)), j=1.. A232626(n)) (mod C(q(2,n), rho(q(2,n)))).
FORMULA
a(n,m) = [x^m] MP2(n, x), n>=1, m = 0, 1, ..., A232626(n), with the minimal polynomials of 2*sin(4*Pi/n), computed like explained above in a comment.
EXAMPLE
The table a(n,m) begins:
--------------------------------------------------------------------------------------
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
1: 0 1
2: 0 1
3: -3 0 1
4: 0 1
5: 5 0 -5 0 1
6: -3 0 1
7: -7 0 14 0 -7 0 1
8: -2 1
9: -3 0 9 0 -6 0 1
10: 5 0 -5 0 1
11: -11 0 55 0 -77 0 44 0 -11 0 1
12: -3 0 1
13: 13 0 -91 0 182 0 -156 0 65 0 -13 0 1
14: -7 0 14 0 -7 0 1
15: 1 0 -8 0 14 0 -7 0 1
16: -2 0 1
17: 17 0 -204 0 714 0 -1122 0 935 0 -442 0 119 0 -17 0 1
18: -3 0 9 0 -6 0 1
19: -19 0 285 0 -1254 0 2508 0 -2717 0 1729 0 -665 0 152 0 -19 0 1
20: 5 0 -5 0 1
...
n=1: 2*sin(4*Pi/1) = 0 is rational, therefore MP2(1, x) = x, with coefficients 0, 1, and degree A232626(1) = 1. PB2(1, rho(1,1)) = PB2(1, rho(1)) = 0. n=3: A232626(2) = 2. PB2(2, x) = -x, C(6, x) = x^2 - 3, with zeros rho(6) and R(5, rho(6)) (for R see A127672), hence rho(6,1) = rho(6) and rho(6,2) = R(5, rho(6))= 5*rho(6) - 5*rho(6)^3 + 1*rho(6)^5, MP2(3, x) = (x - (-rho(6)))*(x - (- R(5, rho(6))) reduced with rho(6)^2 = 3 leading to MP2(3, x) = -3 + x^2, yielding row n=3: -3 0 1. n=8: this row -2, 1 coincides with row n=4 of A231188. n=17: coincides with WolframAlpha's MinimalPolynomial[2*sin(4*Pi/17),x] = 17-204 x^2+714 x^4-1122 x^6+935 x^8-442 x^10+119 x^12-17 x^14+x^16.
Discriminants of the minimal polynomials of 2*sin(2*Pi/n) for n >= 1.
+10
0
1, 1, 12, 1, 2000, 12, 1075648, 8, 1259712, 2000, 2414538435584, 1, 7340688973975552, 1075648, 324000000, 2048, 187591757103747287810048, 1259712, 1436650532447139184230793216, 5, 843466573910016, 2414538435584
COMMENTS
The coefficient list for the minimal polynomials of 2*sin(2*Pi/n), called here MP(1; n, x), is given as A231188.
FORMULA
a(n) = discriminant of MP(1; n, x) = sum( A231188(n,m)*x^m, m=0..deg(1; n)) with the degree deg(1; n) = A093819(n), n >= 1.
EXAMPLE
n=5: MP(1; 5, x) = 5 - 5*x^2 + x^4 with the four zeros x[1] = +sqrt(2 + tau), x[2] = -sqrt(2 + tau), x[3] = +sqrt(3 - tau), x[4] = -sqrt(3 - tau), with the golden section tau := (1 + sqrt(5))/2. They produce the discriminant(MP(1; 5, x)) = (Det(Vandermonde(4,[x[1],x[2],x[3],x[4]]))^2 = (20*sqrt(5))^2 = 2000.
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