It turns out to be the case that the criteria for congruence and representation of numbers by forms with rational coefficients can be reduced to consideration of like problems for forms with*p*-adic coefficients. As was stated in the introduction, the results of this chapter could be stated largely as Minkowski gave them in terms of congruences. Pall has done just this. But since the gain in their introduction seems to justify such treatment, we prove our results in terms of*p*-adic numbers.

For*p*a prime number, we define a*p-adic number*to be any formal series

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