Recover polynomial from 1 root

[Albert Chan]

FYI, this is what manual "rinse and repeat" method look like:

x = r = sqrt(6) / (5^(1/3) + sqrt(3))

3*sqrt(2) x + 5400^(1/6) x = 6 <-- multiply 6/r, both side
5400^(1/6) x = 6 - 3*sqrt(2) x
5400 x^6 = (6 - 3*sqrt(2) x)^6 <-- only square root remains ...

Expand above, and group sqrt(2) terms, we get

sqrt(2) * (81 x^5 + 540 x^3 + 324 x) = x^6 + 405 x^4 + 810 x^2 + 108

Square both side, all radicals are gone. We get the polynomial:

x^12 - 12312 x^10 - 9315 x^8 - 31860 x^6 + 43740 x^4 - 34992 x^2 + 11664 = 0