Reduction of cyclotomic units at the ramified prime ($u_k=({\zeta }_p^k-1)/({\zeta }_p-1)⟼k\mathrm{mod}p$)

For $1\le k<p$, choose $l$ with $kl\equiv 1(\mathrm{mod}p)$. The geometric sum $1+{\zeta }_p^k+\cdots +{\zeta }_p^{k(l-1)}$ is the inverse of $u_k$. Hence these are units, and their residues cover ${\mathbb{F}}_p^{\times }$. In the cyclotomic Milnor square, this makes the $K_1$ map onto the common field surjective and eliminates its connecting obstruction to $K_0$.