Re: Unsolvable polynomial with real roots only-

Przemyslaw Koprowski wrote:
Hi all,

Does anyone knows an example of an unsolvable polynomial of odd degree that have *only real* roots?

To be precise: I'm searching for a polynomial f such that:
1) f has rational coefficients and
2) f has odd degree and
3) f has only real roots and
4) the roots of f cannot be expresed in terms of radicals.

Thanks in advance,
Przemek

P.S.
The "classical" examples do not work:
t^5-t-1 ==> only one real root and two pairs of conjugated nonreal
t^5-10t+2 ==> three real roots and two conjugated nonreal


Try X^5 - 30X^3 + 360X + C; C being a constant <> 0.
this is C + 720 times the 5th-degree McLaurin approximation to (1 - cosX) / X.

Motivation for this choice: Power series for sinX, cosX and closely related functions show enough undulations to fit your purpose.

I did not elaborate further on this. Perhaps one has to add small disturbing terms in order to get indeed five real zeros. Should not be too difficult; just use Maple or perhaps a handheld graphic calculator to explore this.

Then the real guesswork: quintic polynomials typically have S5 as their Galois groups.
Conjecture: the probability to hit upon a quintic with a solvable Galois group by chance is slim.
So the polynomial mentioned or otherwise a perturbation may very well answer your question at first effort.

Ciao: Johan E. Mebius
.

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