# Field theory question

#### ElDavidas

1. Homework Statement

Show $Q(\sqrt{p},\sqrt{q}) = Q(\sqrt{p} + \sqrt{q})$

2. Homework Equations

$p$ and $q$ are two different prime numbers

3. The Attempt at a Solution

I can show $\sqrt{p} + \sqrt{q} \in Q(\sqrt{p},\sqrt{p})$

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

So far I've let $\alpha = \sqrt{p} + \sqrt{q}$

and found the powers $\alpha^2 = p + q + 2 \sqrt{p}\sqrt{q}$ and $\alpha^3 = (p + 3q )\sqrt{p} + (3p + q)\sqrt{q}$

Not sure what to do now though.

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#### matt grime

Homework Helper
So you know that
$\sqrt{p} + \sqrt{q}$

and

$(p + 3q )\sqrt{p} + (3p + q)\sqrt{q}$

are in the field. That is all you need. HINT: think simlutaneous linear equations.