Rings of integers in number fields and root lattices

This paper investigates whether a root lattice can be similar to the lattice $\mathscr{O}$ of all integer elements of a number field $K$ endowed with the inner product $(x,y):=\operatorname{Trace}_{K/\mathbb{Q}}(x\cdot\theta(y))$, where $\theta$ is an involution of the field $K$. For each of the following three properties (1), (2), (3), a classification of all the pairs $K$, $\theta$ with this property is obtained: (1) $\mathscr{O}$ is a root lattice; (2) $\mathscr{O}$ is similar to an even root lattice; (3) $\mathscr{O}$ is similar to the lattice $\mathbb{Z}^{[K:\mathbb{Q}]}$. The necessary conditions for similarity of $\mathscr{O}$ to a root lattice of other types are also obtained. It is proved that $\mathscr{O}$ cannot be similar to a positive definite even unimodular lattice of rank $\le48$, in particular, to the Leech lattice.