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**Deveno** well, the elements 1, √a+√b, (√a+√b)^2 = (a+b) + 2√a√b, (√a+√b)^3 = (2a+b)√a + (a+2b)√b all have to be linearly independent, since [Q(√a+√b):Q] = 4.

so if we set √a+√b = u, √a = (1/(a-b))(u^3 - (a+2b)u), and √b = (1/(b-a))(u^3 - (2a+b)u) (you might want to double-check my algebra).

this shows that Q(√a,√b) is contained in Q(√a+√b) (note as well, that this gives a different basis for Q(√a,√b):

(it is an interesting exercise to show that if we set √a = x, √b = y, that the set {1,x+y,a+b+xy,(2a+b)x+(a+2b)y} is linearly independent if {1,x,y,xy} is).