Thread: The Golden Section. View Single Post
2004-04-27, 13:38   #17
R.D. Silverman

Nov 2003

11101001001002 Posts

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)) + (1-sqrt(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
sub-field 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.