[SeqFan] Re: Suggestion for a sequence: weights on a circle, roots of unity and Chebyshev's polynomials.

[Antti Karttunen]

I just found this:

http://www.research.att.com/~njas/sequences/A113401

A113401 Algebraic degree of Cos[2Pi/n] for constructible n-gons 
(A003401).    

1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, 8, 8, 8, 16, 8, 16, 16, 
16, 32,
16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 128, 64, 64, 64, 64, 64, 
64, 64,
128, 128, 128, 128, 128, 128, 256, 128, 128, 128, 256, 256, 256, 256, 
512, 256,
256, 256, 256, 256, 256, 512

OFFSET 1,5
KEYWORD nonn,easy,nice
AUTHOR E. W. Weisstein (eric(AT)weisstein.com), Oct 28, 2005

I just wonder, isn't this possible at all to compute/estimate for 
non-constructible
polygons (heptagon, 9-gon, etc.)?
(Some holes in my higher algebra, I guess.) From Weisstein's web page:
"The angles m*Pi/n (with m,n integers) for which the trigonometric functions
may be expressed in terms of finite root extraction of real numbers are
limited to values of m which are precisely those which produce 
constructible polygons."
Still I wonder, how much we can say about the vertices of heptagon
(and their algebraic degree) just by using cos(2Pi/7) as x, and then 
applying
Chebyshev polynomials (or some other trick).


Yours,

Antti


I wrote:

>
> The question is this:
>
> What is the (minimal) dimension of a linearly independent base
> (consisting of reals) such that the real-components of all the nth
> roots of unity (i.e. cosines of the corresponding angles) can be
> represented as linear combinations (with rational coefficients) of 
> that base?
>
> E.g. for n=3, i.e. an equilateral triangle whose one vertex is located 
> at (1,0)
> the other two vertices are located at (-(1/2),sqrt(3)/2)
> and (-(1/2),-sqrt(3)/2), the dimension is 1 as
> both values 1 and -1/2 can be represented as the rational
> multiples of 1.
>
> For the 5th roots of unity (regular pentagon) we have the following
> values for real-part: 1, (sqrt(5)-1)/4, -(sqrt(5)+1)/4, 
> -(sqrt(5)+1)/4, (sqrt(5)-1)/4.
>
> So for example 1 and sqrt(5)-1 can be used as a base, and its 
> dimension is 2.
> (Note that sqrt(5)+1 = 1*(sqrt(5)-1) + 2*1.)
>