Can You Prove the Equality of Field Theories with Different Prime Numbers?

[ElDavidas]

Homework Statement

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

Homework Equations

p and q are two different prime numbers

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.

 

[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.