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Field theory question

  • Thread starter ElDavidas
  • Start date
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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.
 

matt grime

Science Advisor
Homework Helper
9,394
3
So you know that
[itex] \sqrt{p} + \sqrt{q}[/itex]

and

[itex] (p + 3q )\sqrt{p} + (3p + q)\sqrt{q}[/itex]

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

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