# Field theory question

1. Jan 9, 2007

### ElDavidas

1. The problem statement, all variables and given/known data

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

2. Relevant 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.

2. Jan 9, 2007

### matt grime

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.