Re: quasi-cyclotomic polynomials

On Sat, 25 Aug 2007 01:53:03 +0100, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

On Fri, 24 Aug 2007 19:35:49 -0400, quasi
<quasi@xxxxxxxx> wrote:

For this discussion, all polynomials will be elements of Z[x].

Call a polynomial "quasi-cyclotomic" if all of its roots have complex
modulus equal to 1.

[...]

Question (1):

Does there exist a nonconstant, primitive, quasi-cyclotomic polynomial
whose leading coefficient is not 1 or -1?

Note -- by primitive, we mean that the coefficients have no common
factor (in Z).

Question (2):

Does there exist a nonconstant, monic, irreducible quasi-cyclotomic
polynomial which is not cyclotomic?

Note -- if the answer to question (2) is "no", then the answer to
question (1) is automatically also "no".

I don't know anything about this myself, but apparently
there's a theorem of Kronecker which says that all the
roots of a quasi-cyclotomic polynomial are roots of unity:

<http://www.groupsrv.com/science/about89561.html>

Hence, the polynomial divides x^n - 1, for some n; this is
a product of cyclotomic polynomials, which are irreducible;
hence it is itself cyclotomic; so the answer to (2) is "no"
(unless I've boobed as usual).

No, the link is fine -- it clearly deals with exact questions I asked.

Thus, according to the link, question (2) has a "no" answer, based on
a theorem of Kronecker.

Also, it appears I made an elementary mistake in my assertion that a
"no" answer to question (2) implies a "no" answer to question (1). In
fact, the answer to question (1) is "yes", as is shown by the easy
example, 5x^2 - 6x + 5, which was given in the link.

Thanks.

quasi
.

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