(1+sqrt(2)+sqrt(3))(1-sqrt(2)-sqrt(3))=-4-2sqrt(6)

so you'd have to multiply again, this time by -4+2sqrt(6)

in general the "conjugate" of a+bX+cY+.... will always be a-bX-cY-.... (similar to how complex numbers work)

where a,b,c,... are from the base field and X,Y,... are the algebraic numbers you extend the field by.