On the trace of the integers of a number field

TL;DR

This paper characterizes when the trace of the ring of integers in a number field is a proper subset of the integers, linking it to prime factors of the degree and ramification, and explores conditions for equality.

Contribution

It provides a new criterion relating the trace of integers in a number field to prime ramification and degree factors, and examines cases of equality for compositum fields.

Findings

Trace is a proper subset of integers if a prime divides all ramification exponents.

Trace equals integers when the field is a compositum of certain number fields.

Characterizes ramification conditions affecting the trace of integers.

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 $Tr(\mathcal{O}_{L})\varsubsetneq \mathbb{Z}$ 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.