CHAPTER IV - Forms with coefficients in R(p)

Summary

Equivalence. We further restrict our forms in this chapter by requiring that their coefficients be p-adic integers, that is, in R(p). As the theory for F(p) led to results for forms with rational coefficients, so the theory for R(p) leads to results for forms with integer coefficients though, as we shall see, there is here an intermediate case which has no previous analogy. Suppose a transformation T with elements in R(p) takes a form f into a form g where both forms are in R(p) and have non-zero determinants d_f and d_g, respectively, and a transformation S in R(p) takes g into f. Then

|T²|·d_f = d_g and |S²|·d_g = d_f

imply that |T|²|S|² = 1. Since the determinants of T and S are p-adic integers, they must be units. Thus, following section 20, we call a transformation unimodular in R(p) or p-adically unimodular if its elements are in R(p) and its determinant a p-adic unit. Two forms are equivalent in R(p) or p-adically equivalent if one may be taken into the other by a unimodular transformation in R(p). Two p-adically equivalent forms represent the same p-adic integers for values of the variables in R(p). In this chapter, unless it is stated to the contrary, equivalence means p-adic equivalence and we denote it by the sign ≅. Two forms equivalent in R(p) are said to be in the same p-adic class.