Let $p$ be an odd prime number and $\zeta_{ p} := \exp(2\pi i/p)$. Then, it is well-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 generated by $(1-\zeta_{ p})^{ -(p-3)/2}$. In this paper, in contrast with the case of the $A_{ p-1}$-root lattice, we prove the following theorem: Let $n$ be an odd positive integer and $F/\mathbb{ Q}$ be a Galois extension of degree $n$. Then, there exist no fractional ideals $\Lambda$ of $F$ such that the restricted trace 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 of fractional 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 cubic lattices) with normal $\mathbb{ Z}$-bases. The latter fact is in contrast with another fact that among quadratic fields only $\mathbb{ Q}(\sqrt{ \pm3})$ contain sub $\mathbb{ Z}$-lattices of type $A_{ 2}$.
Categories: math.NT