Polynomials with roots cos(2*k*pi/n),
where n is an odd number.
by 'Literka'.
1. Introduction.
Let n be a prime number bigger than 2 and u=2*k*pi/n, where k is from the set {1,2,3...(n-1)/2}.
Let pi=3.14159...
Formula of DeMoivre and simple computations show that
Taking
and using the well known formula that sum of squares of sin() and cos() is 1, we receive a polynomial
with roots, which are squares of cos(2*k*pi/n). For example, for n=5, this polynomial is
We can use the formula
to receive another polynomial with roots cos(2*k*pi/n). Just introduce new unknown variable y by the
equation x=(1+y)/2. For example the polynomial
has roots cos(2*pi/5) and cos(4*pi/5).
Notice 1: All these polynomials have degrees (n-1)/2. For example, when n=7, the roots of considered
polynomial are cos(2*pi/7), cos(4*pi/7), cos(8*pi/7). This is the same as to say they are
cos(2*pi/7), -cos(3*pi/7), -cos(pi/7).
Notice 2.
Compute the roots of the previous 2 polynomials. Some of them must be squares of others from another polynomial.
Do you see this?
Notice 3. We assumed that n is a prime number. Similar polynomials can be constructed
for other odd numbers. However for constructions of polynomials Literka used procedures valid only for prime
numbers. For a long time Literka forgot about the assumption n is prime and still it is not known if
this assumption is necessary. Literka
checked validity of constructed polynomials plugging cosine values for x and comparing values of a polynomial
with 0. Later Literka checked
other odd numbers. For all checked numbers polynomials constructed using Literka's procedures are the same
as polynomials received using DeMoivre's formula.
To see other examples of polynomials click download.
2. Few examples.
Let us present polynomials with roots cos(2*k*pi/n). For n=7 this polynomial is
For n=13
And for n=23
3. Fermat numbers.
See a page Angles k*pi/257.
for definition and remarks about Fermat numbers. We have already shown an example with a Fermat number,
namely with a number n=5. Next Fermat number is n=17 and the corresponding polynomial is
Let us notice that this polynomial looks to be much simpler than the polynomial from the page of Literka
Factorization of a polynomial.
Finally, n=257 and a polynomial because of which the current page was created:
This polynomial looks to be complicated and unreadable. However there are some regularities. For example,
denominators are powers of 2. Seeing this polynomial it is hard to imagine how we can factorize it, i.e. to
write as a product of 2 polynomials of 64 degree. But we know the roots of this polynomial and because
of this, it is possible to factorize it. Factorizations of polynomials of this kind are shown on the page of Literka
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
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