Additive coboundaries of 𝔽ₚ(χ) versus μₚ-coboundaries

Let $K$ and $L$ be fields with $L$ a $K$-algebra, let $p$ be a prime, and write $G = L \simeq_{\mathrm{alg}[K]} L$ for the group of $K$-algebra automorphisms of $L$. Let $\chi : G \to (\mathbb{Z}/p)^\times$ be a group homomorphism, and let $\zeta \in L^\times$ be a primitive $p$-th root of unity such that $g \cdot \zeta = \zeta^{\,v(\chi(g))}$ for every $g \in G$, where $v(x) \in \mathbb{N}$ denotes the canonical representative of $x \in \mathbb{Z}/p$. Let $c : G \to \mathbb{Z}/p$ be any function. The assertion is an equivalence: $c$ lies in coboundaries₁ (ofChar χ), that is, $c$ is a $1$-coboundary for the one-dimensional representation of $G$ over $\mathbb{Z}/p$ obtained by twisting the trivial representation on $\mathbb{Z}/p$ by $\chi$ (so $g$ acts as multiplication by $\chi(g)$), if and only if there exists a unit $\eta \in L^\times$ with $\eta^p = 1$ and $g \cdot \eta / \eta = \zeta^{\,v(c(g))}$ for all $g \in G$.

This is the coboundary half of the dictionary between $\mathbb{F}_p$-valued cohomology of the character $\chi$ and multiplicative Kummer-theoretic cocycles in $\mu_p \subset L^\times$; together with the corresponding statement for cocycles it matches classes in $H^1(G,\mu_p)$ with classes in $H^1(G,\mathbb{F}_p(\chi))$. It is used in the counting of continuous classes for ofChar χ and in the bound on the rank of cocycles attached to the cyclotomic character on unit inertia.