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## Root lattices in number fields

Bulletin of Mathematical Sciences. 2021. Vol. 11. No. 3. P. 2050021-1-2050021-22.

Popov Vladimir L., Zarhin Y.

We explore whether a root lattice may be similar to the lattice O of integers of a

number field K endowed with the inner product (x, y) := TraceK/Q(x · θ(y)), where θ is

an involution of K. We classify all pairs K, θ such that O is similar to either an even

root lattice or the root lattice Z^[K:Q]. We also classify all pairs K, θ such that O is a

root lattice. In addition to this, we show that O is never similar to a positive-definite

even unimodular lattice of rank ≤ 48, in particular, O 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.