Level-constant classes in H¹(χ) count K^×/(K^×)ᵖ

Let $K\subseteq L$ be fields with $L/K$ Galois, let $p$ be a prime, write $G=L\simeq_{\mathrm{alg}[K]}L$ for the group of $K$-algebra automorphisms of $L$, and let $\chi\colon G\to(\mathbb{Z}/p)^\times$ be a group homomorphism. Here ofChar χ denotes the one-dimensional representation of $G$ over $\mathbb{Z}/p$ given by $g\mapsto \chi(g)\cdot\mathrm{id}$, i.e. the trivial representation on $\mathbb{Z}/p$ twisted by $\chi$. Assume given $\zeta\in L^\times$, a primitive $p$-th root of unity, such that $g\cdot\zeta=\zeta^{(\chi g).\mathrm{val}}$ for every $g\in G$, and assume that every $a\in K^\times$ acquires a $p$-th root in $L^\times$, i.e. for each $a$ there is $\alpha\in L^\times$ with $\mathrm{algebraMap}_{K,L}(a)=\alpha^p$. Let adm be a $\mathbb{Z}/p$-submodule of $H^1(\mathrm{ofChar}\,\chi)$ whose elements are characterised as follows: $x\in$ adm if and only if $x$ is the image under the canonical projection H1π of some $1$-cocycle $c$ for which there exists an intermediate field $E$ of $L/K$, finite-dimensional over $K$, with $c(g s)=c(g)$ for all $g\in G$ and all $s$ in the fixing subgroup of $E$. Then the cardinality of adm equals the cardinality of $K^\times$ modulo the image of the $p$-th power homomorphism powMonoidHom p on $K^\times$, both computed as Nat.card (so the common value is $0$ if either side is infinite).

This is Kummer theory in the form adapted to the full (possibly infinite) Galois group: the subgroup of classes in $H^1(G,\mu_p)$ represented by a cocycle constant on cosets of $\mathrm{Gal}(L/E)$ for some finite subextension $E/K$ — the classes of finite level — is in bijection with $K^\times/(K^\times)^p$. It is used in the computation of the local terms entering the dual Selmer group estimate, being cited in the specialisation of this count to the cyclotomic character at a prime.