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Old 2004-04-27, 15:02   #22
R.D. Silverman
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Nov 2003

11101001001002 Posts

Originally Posted by jinydu
I think there was somewhat of a misunderstanding. This is what I meant:

x = cbrt(10 + sqrt(108)) + cbrt(10 - sqrt(108)),

just as I would be unsatisfied with saying that the solution to x^2 - 2x + 1 = 0 is

x = (-(-2)+-sqrt((2^2)-(4*1*1)))/(2*1).

As in the quadratic example, I would like to go through a series of well-defined steps to arrive at the solution

x = 2.
I gave the steps. If the expression can be simplified it must lie in the
ground field (Q) or in some sub-field of the full splitting field of your
polynomial. If it is in the ground field then a quick, rough, numerical
approximation to the root will tell you the answer. If it is in a subfield you
must first find a basis for that sub-field, express the elements of the
subfield as a linear combination of basis elements, equate to your root
and solve for the coefficients. The coefficients will be integers and
can be found by a number of methods. Look up the "relation finding"
algorithm of Ferguson & Forcade.

What more do you want? If you want the details of how to compute the
sub-fields and their bases you will need to learn some algebraic number theory.
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