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

In this article, we consider integral trace forms of cyclotomic fields given by the restrictions of trace forms TrQ(ζn)/QTrQ(ζn)/Q to OQ(ζn)OQ(ζn) and we get orthogonal decompositions of symmetric ZZ-bilinear modules OQ(ζn)OQ(ζn) into rank one and rank two symmetric ZZ-bilinear modules in an explicit way. As a result, we can get canonical forms of symmetric ZpZp-bilinear modules Zp⊗ZOQ(ζn)Zp⊗ZOQ(ζn) over ZpZp the ring of p-adic integers.