Re: Proof there exists another system of polynomials like X^n-1 with only real roots.
- From: "Mariano Suárez-Alvarez" <mariano.suarezalvarez@xxxxxxxxx>
- Date: Tue, 22 Jan 2008 07:09:03 -0800 (PST)
On Jan 22, 8:14 am, Gerry <Gerry...@xxxxxxxxx> wrote:
Hi all,
X^n-1=0 splits into the cyclotomic polynomials and the only roots
which are not complex are 1,-1.
Which other equivalent system of polynomials splits into the same
number of irreducible polynomials as X^n-1=0 does and has apart from
the root 1 only real roots?
Take any sequence (p_n : n >= 1) of polynomials such that
* p_n is irreducible for all n, with all its roots real;
* for all n and m, p_n and p_m do not have any common
roots;
* p_1 = X;
* for extra niceness, suppose that deg p_n = phi(n),
the Euler phi function.
Now consider the polynomials
q_n = product_{d divides n} p_d,
where the product is taken over all divisors 1 <= d <= n of n.
This sequences does what you want.
In fact, this procedure gives all examples.
-- m
.
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