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I need to prove that any rational number a/b can be expressed as the sum of 4 squares of rational numbers

I have tried something similar to the proof for integers, but can't get it to work. I have tried an algebraic proof, but I keep running into the issie of potential irrational denominators

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closed as off-topic by Namaste, max_zorn, Brahadeesh, Ali Caglayan, Christopher Nov 29 '18 at 10:02

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  • $\begingroup$ So I assume you mean the sum of 4 squares of rational numbers? $\endgroup$ – JDMan4444 Nov 29 '18 at 0:15
  • $\begingroup$ yes, thank you. $\endgroup$ – user2782067 Nov 29 '18 at 0:18
  • $\begingroup$ I don't understand, you can take $\frac{c}{d}$ ,$\frac{e}{f}$ , $\frac{g}{h}$ and $\frac{i}{j}$ square them and sum $\endgroup$ – Enigsis Nov 29 '18 at 0:22
  • $\begingroup$ @Enigsis It is the issue of showing, say, $1/7$ is equal to such a sum. $\endgroup$ – JDMan4444 Nov 29 '18 at 0:23
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You are taking $\frac{a}{b}$ with integers $a,b > 0.$ Thus $ab$ is a positive integer.

Let $$ w^2 + x^2 + y^2 + z^2 = ab $$

Then $$ \left( \frac{w}{b} \right)^2 + \left( \frac{x}{b} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{b} \right)^2 = \frac{a}{b}$$

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