# Trace Forms of Certain Subfields of Cyclotomic Fields and Applications

### Abstract

In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9,Λ10\Lambda_9, \Lambda_{10}Λ9,Λ10 and Λ11\Lambda_{11}Λ11 and they are images of Z\mathbb{Z}Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 777 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.

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