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

[Gene Smith]

On 5/1/06, Antti Karttunen <antti.karttunen at gmail.com> wrote:
>
>
> For example, can the vertices of a regular heptagon given exactly
> without resorting to trigonometric functions?



This involves the maximal real subfield of the roots of unity,
w=z+1/z where z is a root of unity. These are cyclic extensions, so all of
the roots can be expressed in terms of powers of one of them. Trig functions
can be used but are not essential. For seventh roots of unity, regular
heptagons, you get w^3+w^2-2w-1=0, an irreducible cubic which is why you
can't construct the heptagon with ruler and compass. If w is one root, the
others are w^2-2 and 1-w-w^2.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20060503/2da36b49/attachment-0001.htm>