Continuous H¹(ℚₚ,μₚ) counted by ℚₚ^×/(ℚₚ^×)ᵖ

Fix a prime $p$ and a prime $q$ with $q = p$ as natural numbers. Let $\Gamma_q$ denote primeLocalGaloisGroup q, the group of $\mathbb{Q}_q$-algebra automorphisms of the algebraic closure PadicAlgCl q, and let primeLocalToGlobal q be the homomorphism $\Gamma_q \to \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ obtained by restricting scalars to $\mathbb{Q}$ and then restricting to the normal subextension $\overline{\mathbb{Q}}$. Consider the one-dimensional representation ofChar of $\Gamma_q$ over $\mathbb{Z}/p$ attached to the character $(\mathrm{cycloChar}\ p)\circ(\mathrm{primeLocalToGlobal}\ q)$, that is, the trivial representation on $\mathbb{Z}/p$ twisted so that $g$ acts by the mod $p$ cyclotomic character of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ evaluated at the image of $g$. Let $\mathrm{adm}_1$ be a $\mathbb{Z}/p$-submodule of the first cohomology $H^1$ of this representation whose members are exactly the classes $x$ admitting a $1$-cocycle $c$ representing $x$ (under the projection H1π from cocycles to $H^1$) for which there is an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, with $c(gs) = c(g)$ for all $g, s \in \Gamma_q$ such that the image of $s$ fixes $F$ pointwise. Then the cardinality of $\mathrm{adm}_1$ equals that of $\mathbb{Q}_p^\times$ modulo the image of the $p$-th power homomorphism, i.e. of $\mathbb{Q}_p^\times/(\mathbb{Q}_p^\times)^p$.

This is the local Kummer-theoretic count of the continuous (level-constant) part of $H^1(\mathbb{Q}_p,\mu_p)$, the submodule of classes trivialised on the Galois group of a finite layer coming from a number field. It feeds the local Euler-characteristic bookkeeping: it is used by groupCohomology.finrank_continuousClasses_ofChar_cycloChar_eq_two_of_primeLocal, which converts this cardinality into the dimension statement $\dim_{\mathbb{F}_p} = 2$ for odd $p$.