Norm projection formula for cyclic Brauer pairings ($\mathrm{Cor}(\mathrm{Res}\chi ,a)=(\chi ,N_{L/K}a)$)

For a finite extension $L/K$, an unramified or general cyclic character $\chi$ over $K$, and $a\in L^{\times }$, transfer satisfies ${\mathrm{Cor}}_{L/K}(\mathrm{Res}\chi ,a)=(\chi ,N_{L/K}a)$. It is the cup-product projection formula combined with the field norm on multiplicative degree-zero coefficients. With the norm definition of Brauer transfer it also covers purely inseparable steps.