On the trace of the integers of a number field

Abstract

Let $Tr$ denote the trace $\mathbb{Z}$-module homomorphism defined on the ring ${\mathcal{O}}_L$ of the integers of a number field $L.$ We show that $Tr({\mathcal{O}}_L)\mathbb{Z}$ if and only if there is a prime factor $p$ of the degree of $L$ such that if ${\wp }_1^{e_1}...{\wp }_s^{e_s}$ is the prime factorization of the ideal $p{\mathcal{O}}_L$ in ${\mathcal{O}}_L,$ then $p$ divides all powers $e_1,...,e_s.$ Also, we prove that the equality $Tr({\mathcal{O}}_L)=\mathbb{Z}$ holds when $L$ is the compositum of certain number fields.

6 pages