sage: R.<x> = QQ[]
sage: K.<a,b> = NumberField( [x^3-2, x^2+3])
sage: a^3 == 2, b^2 == -3
(True, True)
sage: a.absolute_minpoly(), b.absolute_minpoly()
(x^3 - 2, x^2 + 3)
sage: u = (-1+b)/2    # primitive 3.rd root of unity
sage: u^3
1
sage: [ (a*u^k).absolute_minpoly() for k in [0,1,2] ]
[x^3 - 2, x^3 - 2, x^3 - 2]
sage: (a^2+b^3).absolute_minpoly()
x^6 + 81*x^4 - 8*x^3 + 2187*x^2 + 648*x + 19699

sage: S.<Y> = K[]
sage: S
Univariate Polynomial Ring in Y over Number Field in a with defining polynomial x^3 - 2 over its base field
sage: prod( [ Y-(a*u^k)^2-(b*sgn)^3 for k in [0,1,2] for sgn in [-1,1] ] )
Y^6 + 81*Y^4 - 8*Y^3 + 2187*Y^2 + 648*Y + 19699

sage: prod( [ Y-(a*u^k)^2-b^3 for k in [0,1,2] ] )
Y^3 + 9*b*Y^2 - 81*Y - 81*b - 4

sage: prod( [ Y-a^2-(b*sgn)^3 for sgn in [-1,1] ] )
Y^2 - 2*a^2*Y + 2*a + 27