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

TL;DR

This paper proves that for odd degree Galois extensions of the rationals, the trace form cannot be of types A, D, or E, contrasting with known cases for cyclotomic fields, and explores lattice structures in cubic fields.

Contribution

It establishes the non-existence of fractional ideals with trace forms of types A, D, E in odd degree Galois extensions, and shows the abundance of type A lattices in cyclic cubic fields.

Findings

No fractional ideals yield trace forms of types A, D, E in odd degree Galois extensions.

Every cyclic cubic field contains infinitely many sub lattices of type A.

Quadratic fields only contain certain type A lattices in specific cases.

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 $\mathbb{Q}(\sqrt{\pm3})$ contain sub $\mathbb{Z}$-lattices of type $A_{2}$.