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**1. Homework Statement**

Show [itex]Q(\sqrt{p},\sqrt{q}) = Q(\sqrt{p} + \sqrt{q}) [/itex]

**2. Homework Equations**

[itex]p[/itex] and [itex]q[/itex] are two different prime numbers

**3. The Attempt at a Solution**

I can show [itex]\sqrt{p} + \sqrt{q} \in Q(\sqrt{p},\sqrt{p}) [/itex]

I have trouble with the other direction though, i.e [itex]\sqrt{p},\sqrt{p} \in Q(\sqrt{p} + \sqrt{q}) [/itex].

So far I've let [itex] \alpha = \sqrt{p} + \sqrt{q}[/itex]

and found the powers [itex] \alpha^2 = p + q + 2 \sqrt{p}\sqrt{q}[/itex] and [itex] \alpha^3 = (p + 3q )\sqrt{p} + (3p + q)\sqrt{q}[/itex]

Not sure what to do now though.