Quote:
Originally Posted by jinydu
Ok, here is what I mean by simplify:
simple:integer>rational number>sqrt or cbrt of a rational number > cbrt(a+sqrt(b)), where a and b are rational numbers:complicated
Thus, cbrt(10 + sqrt(108)) + cbrt(10  sqrt(108)) is complicated.
(1+sqrt(3)) + (1sqrt(3)) is simpler
cbrt(8) is simpler still
2/1 is even simpler
2 is the simplest

You have not given a rigorous, meaningful definition. You need to give
a rigorous metric for 'complexity of an expression'.
In any event, if you mean simplify into one of the forms:
integer
rational number
root of a rational number
You do it the way I outlined.
If you mean: simplify to the form (a + b sqrt(c)) it is clear that it can not
be done. The cubic extension Q(alpha) where alpha is a root of the original
equation has no such quadratic sub field.
The forms to which an expression alpha = cbr(a + sqrt(b)) + cbr(a  sqrt(c))
can be simplified are sharply limited. Any such form must lie within a
subfield of Q(alpha).
To understand what forms are possible you need to understand some Galois
theory/algebraic number theory. Since the polynomial is monic its root
is an element of the maximal order of Q(alpha). In asking whether there
is a simpler form that is not in the ground field (Q in this case) you are
partially asking for separable extensions. See, for example Algorithmic
Algebraic Number Theory by Pohst & Zassenhaus, or H. Cohen's book.
My earlier example shows how to determine if the expression reduces
to an element in the ground field.