= Trace reduction in a ramified local extension
{title2=$\overline{\operatorname{Tr}_{L/F}(a)}=e\operatorname{Tr}_{\kappa_L/\kappa_F}(\bar a)$}
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.
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