AP
Posts:
137
Registered:
3/4/09


Re: Cubics and roots
Posted:
Oct 16, 2012 2:33 AM


On Mon, 15 Oct 2012 09:58:58 +0100, Robin Chapman <R.J.Chapman@ex.ac.uk> wrote:
>On 14/10/2012 13:45, Bill Taylor wrote: >> The heptagon/trisection thread got me thinking >> about an old query of mine, concerning the reach >> of *real* cube roots, (NOT complex). >> >> Am I correct in thinking: that a cubic with >> rational coefficients but no rational roots, >> has solutions obtainable(from 1) via the four >> basic operations AND real cube roots, >> ** ifandonlyif ** >> it has precisely one real root? > >casus irreducibilis
ex : P=X^33X+1 is irreducible on Q
his roots are 2cos(pi/9) ; 2cos(7pi/9) ; 2cos(5pi/9) not obtainable via the four basic operations and squre, cube,.. roots of positifs numbers
but P=X^315X4 is reducible and his roots are 4 ;2sqrt(2);2+sqrt(3)
see Otto Hölder , 1891

