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Let $p$ be an odd prime number and $\zeta_{p} := \exp(2\pi i/p)$. Then, it iswell-known that the $A_{p-1}$-root lattice can be realized as the (Hermitian)trace form of the $p$-th cyclotomic extension$\mathbb{Q}(\zeta_{p})/\mathbb{Q}$ restricted to the fractional ideal generatedby $(1-\zeta_{p})^{-(p-3)/2}$. In this paper, in contrast with the case of the$A_{p-1}$-root lattice, we prove the following theorem: Let $n$ be an oddpositive integer and $F/\mathbb{Q}$ be a Galois extension of degree $n$. Then,there exist no fractional ideals $\Lambda$ of $F$ such that the restrictedtrace form $(\Lambda, \mathrm{Tr}|_{\Lambda \times \Lambda})$ is of type$A_{n}, D_{n}, E_{n}$. The proof is done by the prime ideal factorization offractional ideals of $F$ with care of certain 2-adic obstruction. Additionally,we prove that every cyclic cubic field contains infinitely many distinct sub$\mathbb{Z}$-lattices of type $A_{3}$ (i.e., normalized face centered cubiclattices) with normal $\mathbb{Z}$-bases. The latter fact is in contrast withanother fact that among quadratic fields only $\mathbb{Q}(\sqrt{\pm3})$ containsub $\mathbb{Z}$-lattices of type $A_{2}$.

Galois trace forms of type $A_{n}, D_{n}, E_{n}$ for odd $n$