Algebraic to symbolic expression

edit

You can use the radical_expression method:

sage: a = AA(sqrt(3)+sqrt(2))                                                                                                                                                                                
sage: a                                                                                                                                                                                                      
3.146264369941973?
sage: a.radical_expression()                                                                                                                                                                                 
sqrt(2*sqrt(6) + 5)

Note that this is not a "complete" feature, as there might be some algebraic numbers admitting a radical decomposition that this feature will not find.

I'm not sure how minpoly() works but I'm pretty sure the following numerical approach (using PSLQ) is different:

z = AA(sqrt(3)+sqrt(2))
max_degree = 4
from mpmath import mp, findpoly
mp.dps = 15 # decimal places
f = ZZ['x'](findpoly(z.numerical_approx(digits=mp.dps), max_degree))
print(f)
print(f(z.numerical_approx(digits=mp.dps)))
print(f(z).is_zero())
print(f.roots(SR))

Output:

x^4 - 10*x^2 + 1
1.31006316905768e-14
True
[(-sqrt(2*sqrt(6) + 5), 1), (sqrt(2*sqrt(6) + 5), 1), (-sqrt(-2*sqrt(6) + 5), 1), (sqrt(-2*sqrt(6) + 5), 1)]

It can be used to find minimal polynomials with some degree of uncertainty, but likely at a higher speed. I hope it can be useful.

Edit: The uncertainty of z being a root can be eliminated by checking f(z).is_zero(), and the uncertainty of minimality could be eliminated e.g. by checking f.is_irreducible().