question on correct mathematical construct to hold coordinates

[American Citizen <website.reader3@gmail.com>]

Hello all:

My experiment with trying to use number fields failed to keep accurate 
accounting of rationality, upon rotations of coordinates has failed, 
most likely due to my lack of knowledge of what polynomials are 
legitimate or not for initializing the field. For example 
nfinit(x^2-1/2) didn't work too well for me.

I need the proper GP-Pari mathematical construct to hold point 
coordinate data, which starts at a rational vector, say [1/3, 2/3, 2/3] 
and gets rotated by some angle, or by a bivector. The cosine is 
rational, but the sine may or may not be rational, depending if the 
cosine is Pythagorean or not. (2mn/m^2-n^2)

Is there a way to avoid shoving all the point coordinate data after a 
rotation into the t_REAL field and to keep the 2nd or 4th degree 
polynomial, which the points coordinates will lie in? I could use 
algdep(pt[x,y],2) or algdep(pt[x,y],4) to recover the polynomial from 
t_REAL after the rotation, and then know the roots correctly instead of 
trying to make pull from t_REAL --> t_FRAC and not being able to 
distinguish if the t_REAL is irrational (2nd or 4th degree poly) or 
rational?

Has anyone worked with point coordinates this way? I would appreciate 
some tips.

Otherwise I will have to crank the precision up to 1000 digits or so, do 
the rotations, then use bestappr() or contfracpnqn(contfrac()) to 
recover the rationals, if they exist, or hope that algdep(pt,4) knows 
what to do also.

Randall