Galois trace forms of type An,Dn,En 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 $Q\left({\zeta }_p\right)/Q$ 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 $F/Q$ be a Galois extension of degree n. Then, the number field F does not contain a fractional ideal Λ such that the restricted trace form $(\Lambda ,\mathrm{Tr}{|}_{\Lambda \times \Lambda })$ is of type $A_n,D_n,E_n$. In the proof, we use the prime ideal factorization in F with care of certain 2-adic obstruction for Λ being of type $A_n,D_n,E_n$. Additionally, we prove that every cyclic cubic field contains infinitely many lattices of type $A_3$ (i.e., normalized face centered cubic lattices) having normal $Z$-bases. The latter fact is in contrast with another fact that among quadratic fields only $Q\left(\sqrt{\pm 3}\right)$ contain lattices of type $A_2$.