ADDITION : α = √2 and β = √3

aPb=Sqrt[2]+Sqrt[3]

(* Y^2 -3 --> Minimal Polynomial of √2  *)

R=Resultant[(X-Y)^2-2,Y^2-3,Y]

Simplify[R/.X->aPb]

MULTIPLICATION: α=√2+√3 and β=(19^(1/7))

f=Simplify[Y^7((X/Y)^7 -19)]

(* y^7-19 -> Minimal Polynomial of β *)
(* Y^4 - 10 Y^2 +1 -> Minimal Polynomial of α *)

R=Resultant[X^7-19 Y^7,Y^4 - 10 Y^2 +1,Y]

Simplify[R/.X->(aPb * 19^(1/7))]

(* R -> Vanishing Polynomial of αβ *)

LINEAR COMBINATION: α=(19^(1/7)) (√2 + √3) and β=i

nb=(1/5)*aPb*(19)^(1/7)+(13/17)*I

(* R is a vanishing polynomial of α *)
R

Simplify[Resultant[((X-(1/5)Y)/(13/17))^2 + 1,R/.X->Y,Y]]

Simplify[%/.X->nb]

P(X)=X^3 + X + 1,  α=2^{1/3}

beta=2^(2/3)+2^(1/3)+1

P=X^2+X+1

R=Resultant[Y-P,X^3-2,X]

Simplify[R/.Y->beta]