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)/Q to OQ(ζn) and we get orthogonal decompositions of symmetric Z-bilinear modules OQ(ζn) into rank one and rank two symmetric Z-bilinear modules in an explicit way. As a result, we can get canonical forms of symmetric Zp-bilinear modules Zp⊗ZOQ(ζn) over Zp the ring of p-adic integers.