## Every natural number which is a sum of rational squares is a sum of integer squares

Prove that every natural number which is a sum of rational squares is also a sum of integer squares.

Let $N$ be a natural number and suppose $N = \left(\dfrac{a}{b}\right)^2 + \left(\dfrac{c}{d}\right)^2$. Then we have $(bd)^2N = (ad)^2 + (cb)^2$, so that $(bd)^2N$ is a sum of integer squares. Let $q$ be a prime dividing $N$ which is congruent to 3 mod 4; then $q$ divides $(bd)^2N$, and since this number is a sum of integer squares, by Corollary 19 on page 291 of D&F the multiplicity of $q$ in the factorization of $(bd)^2N$ is even. Then the multiplicity of $q$ in the factorization of $N$ is also even. Thus by Corollary 19, $N$ is a sum of integer squares.