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

Abstract

Let $p$ be an odd prime number and $ζ_{p} := \exp(2πi/p)$. 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 $\mathbb{Q}(ζ_{p})/\mathbb{Q}$ restricted to the fractional ideal generated by $(1-ζ_{p})^{-(p-3)/2}$. 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 $Λ$ of $F$ such that the restricted trace form $(Λ, \mathrm{Tr}|_{Λ\times Λ})$ 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 $A_{2}$.