Reduction of cyclotomic units at the ramified prime
= Reduction of cyclotomic units at the ramified prime
{title2=$u_k=(\zeta_p^k-1)/(\zeta_p-1)\longmapsto k\bmod p$}
For $1\le k<p$, choose $l$ with $kl\equiv1\pmod p$. 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 $K_0$.