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

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