Trace reduction in a ramified local extension ($\overline{{\mathrm{Tr}}_{L/F}(a)}=e{\mathrm{Tr}}_{{\kappa }_L/{\kappa }_F}(\bar{a})$)

For an integral element of a finite extension of non-Archimedean local fields, reduce the multiplication endomorphism on ${\mathcal{O}}_L$ modulo a base uniformizer. Its filtration by the powers of the maximal ideal has $e$ successive quotients, each the residue field ${\kappa }_L$. Multiplication by $a$ induces multiplication by $\bar{a}$ on each quotient. Additivity of the matrix trace through the filtration proves the formula. If the residue characteristic divides the ramification index, the trace of every integral element lies in the base maximal ideal; a Galois extension is not required.