Rings of Integers in Number Fields and Root Lattices

Abstract: This paper investigates whether a root lattice can be similar to the lattice θ of all integer elements of a number field K endowed with the inner product (x, y):=Trace_{K /Q}(x.θ(y)), where θ is an involution of the field K. For each of the following three properties (1), (2), (3), a classification of all the pairs K, θ with this property is obtained: (1) θ is a root lattice; (2) θ is similar to an even root lattice; (3) θ is similar to the lattice ℤ[_K:Q]. The necessary conditions for similarity of θ to a root lattice of other types are also obtained. It is proved that θ cannot be similar to a positive definite even unimodular lattice of rank ≤48, in particular, to the Leech lattice.