Root lattices in number fields

We explore whether a root lattice may be similar to the lattice of integers of a number field K endowed with the inner product (x,y):= TraceK/a,s(x a(y)), where is an involution of K. We classify all pairs K, such that is similar to either an even root lattice or the root lattice a,[K:a,s]. We also classify all pairs K, such that is a root lattice. In addition to this, we show that is never similar to a positive-definite even unimodular lattice of rank 48, in particular, 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 O.