Galois trace forms of type $A_n,D_n,E_n$ for odd $n$

Abstract

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 $\mathbb{Q}(\sqrt{\pm 3})$ contain sub $\mathbb{Z}$-lattices of type $A_2$.