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Topic: Cubics and roots
Replies: 24   Last Post: Oct 19, 2012 4:15 PM

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AP

Posts: 137
Registered: 3/4/09
Re: Cubics and roots
Posted: Oct 16, 2012 2:33 AM
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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,
>> ** if-and-only-if **
>> it has precisely one real root?

>
>casus irreducibilis


ex :
P=X^3-3X+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^3-15X-4 is reducible
and his roots
are 4 ;-2-sqrt(2);-2+sqrt(3)

see Otto Hölder , 1891



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