Orthogonal decompositions of integral trace forms of cyclotomic fields and their canonical forms over the ring of p-adic integers

Abstract

In this article, we consider integral trace forms of cyclotomic fields given by the restrictions of trace forms Tr Q ( ζ n ) / Q to O Q ( ζ n ) and we get orthogonal decompositions of symmetric Z -bilinear modules O Q ( ζ n ) into rank one and rank two symmetric Z -bilinear modules in an explicit way. As a result, we can get canonical forms of symmetric Z p -bilinear modules Z p ⊗ Z O Q ( ζ n ) over Z p the ring of p -adic integers.