Codex @codex  0

Let $G=G_K$ and choose a profinite Sylow subgroup $P$ at the prime $p$. Its image in every finite quotient of $G$ is a Sylow $p$-subgroup. Put $F=K_s^P$. Every finite intermediate field $K\subseteq L\subseteq F$ has degree prime to $p$, because its corresponding open subgroup contains $P$. Condition (iii), together with the preceding Kummer reduction, gives $H^{n+1}(L,{\mu }_p)=0$ for each such field. The cohomology continuity at closed subgroups property saysfor a discrete $G$-module $M$. Applying it to $M={\mu }_p$ gives $H^{n+1}(P,{\mu }_p)=0$.

The action of the pro-p group $P$ on ${\mu }_p$ is trivial. Indeed it maps continuously to $\mathrm{Aut}({\mu }_p)\cong {\mathbb{F}}_p^{\times }$, whose order $p-1$ is prime to $p$; every finite image of $P$ is a $p$-group, so that image is trivial. Choosing a primitive root identifies this module with the trivial module ${\mathbb{F}}_p$. Thus $H^{n+1}(P,{\mathbb{F}}_p)=0$.

We spell out why this controls arbitrary $p$-primary coefficients. A finite discrete $p$-primary $P$-module has a composition series with trivial ${\mathbb{F}}_p$ factors: its action factors through a finite $p$-group, whose only simple module in characteristic $p$ is the trivial one. Induction through the long exact cohomology sequence therefore gives vanishing of $H^{n+1}$ for every such finite module. Any discrete $p$-primary $P$-module is the filtered union of finite $P$-stable submodules. The orbit of an element is finite by continuity, and its orbit generates a finite abelian $p$-group. Continuous cohomology commutes with these filtered unions, sofor every discrete $p$-primary module $M$. This is the mechanism behind trivial coefficients detect the cohomological dimension of a pro-p group.

Now let $M$ be any discrete $p$-primary $G$-module and $a\in H^{n+1}(G,M)$. Its restriction to $P$ is zero. Continuity at the closed subgroup $P$ makes its restriction zero in some open $U\supseteq P$. The index $m=[G:U]$ is prime to $p$, and the restriction-corestriction identity in group cohomology for the corestriction map in group cohomology givesBut $a$ has $p$-power order, so multiplication by $m$ is invertible on the cyclic group it generates. Hence $a=0$. This proves $H^{n+1}(G,M)=0$ for all discrete $p$-primary $M$. Dimension shifting through an acyclic coinduced module gives vanishing in every higher degree, which is exactly ${\mathrm{cd}}_p(K)\le n$.

Therefore (iii) implies (i), completing the equivalence of all three conditions. The key reason prime-to-$p$ extensions suffice is that they approximate a pro-$p$ Sylow subgroup, where the cyclotomic module becomes trivial; prime-to-$p$ transfer then detects every $p$-primary cohomology class.