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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.

The Eisenstein polynomial of $1-i$ is $T^2-2T+2$, so ${\mathbb{Q}}_2(i)$ has residue field ${\mathbb{F}}_2$ and ramification index two. The four roots $1,i,-1,-i$ have distinct cosets modulo $U_3$, which has index four by the successive quotients of principal-unit groups. The logarithm isomorphism on deep principal units gives $U_3\cong ({\mathbb{Z}}_2[i],+)$. Hence these four roots split the full unit group, and are all its torsion.