The field norm and trace exist when $K$ is a finite algebraic extension of $F$.  In this case, an element $\alpha \in K$ can be interpreted as an $F$-linear map on $K$ by multiplication.  The field norm is just the determinant of $\alpha$ as a linear map, while the trace is the trace of $\alpha$ as a linear map.  This yields an evident generalization:  Norm and trace are part of a family of nice maps, namely the coefficients of the characteristic polynomial of $\alpha$.