Let $K$ be a number field and $O_K$ be its ring of integers. The trace form of $K$ is a non-degenerate symmetric $\mathbb{Q}$-bilinear form on $K$ defined by the trace map $\mathrm{Tr}_{K/\mathbb{Q}}$ which also defines a symmetric $\mathbb{Z}$-bilinear form on $O_K$, called the integral trace form of $K$. Since 1980s, the importance of trace forms has been widely recognized and the computation of trace forms have been carried out for some important cases to know the structure of them. In this talk, we treat the same problem for integral trace forms and give orthogonal decompositions of integral trace forms of cyclotomic fields and certain trinomial extensions. Moreover, we determine the canonical forms of them over the ring of $p$-adic integers.

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