Discriminants of number fields and surjectivty of trace homomorphism on rings of integers

TL;DR

This paper surveys elementary criteria for the surjectivity of the trace operator on number field rings of integers and introduces a new criterion based on prime factorization and discriminant properties.

Contribution

It provides a concise survey of known criteria and proposes a novel, simple criterion for trace surjectivity depending only on prime factorization and discriminant squarefree part.

Findings

Survey of elementary trace surjectivity criteria

Introduction of a new criterion based on prime factorization and discriminant

Potential simplification in checking trace surjectivity

Abstract

In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity criterion depending only on the prime factorization of the degree $n$ of $K$ and on the squarefree part of the discriminant $d_K$.