Root lattices in number fields

Abstract

We explore whether a root lattice may be similar to the lattice [Formula: see text] of integers of a number field [Formula: see text] endowed with the inner product [Formula: see text], where [Formula: see text] is an involution of [Formula: see text]. We classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is similar to either an even root lattice or the root lattice [Formula: see text]. We also classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is a root lattice. In addition to this, we show that [Formula: see text] is never similar to a positive-definite even unimodular lattice of rank [Formula: see text], in particular, [Formula: see text] is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of [Formula: see text].