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

Let be a natural number and suppose . Then we have , so that is a sum of integer squares. Let be a prime dividing which is congruent to 3 mod 4; then divides , and since this number is a sum of integer squares, by Corollary 19 on page 291 of D&F the multiplicity of in the factorization of is even. Then the multiplicity of in the factorization of is also even. Thus by Corollary 19, is a sum of integer squares.