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.

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