Note on quadratic forms over the rational field

There is perhaps some methodological interest in developing the theory of quadratic forms over the rational field using only the methods of elementary arithmetic. Hitherto it has appeared necessary to use theorems of a fairly deep nature, most often Dirichlet's theorem about the existence of primes in arithmetic progressions (e.g. Minkowski(1), Hasse(2), Dickson(8), Skolem(9), Burton Jones(6)). Skolem(5) uses a weaker form of Dirichlet's theorem which is rather easier to prove and Siegel(4) uses instead the machinery of the Hardy-Littlewood circle method. In this note I indicate how it is possible to develop the theory of quadratic forms over the rationals without using extraneous resources. Pall(10) states that he has also found such a development of the theory but he does not appear to have published it.