Quote:
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 welldefined 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 subfield 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 subfield, 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
subfields and their bases you will need to learn some algebraic number theory.