We first prove the second isomorphism. Let $(\mathcal{L}, \alpha)$ be a pair as above. Choose an affine open covering $S = \bigcup U_i$ such that $\mathcal{L}|_{U_i} \cong \mathcal{O}_{U_i}$. Say $s_i \in \mathcal{L}(U_i)$ is a generator. Then $\alpha(s_i^{\otimes n}) = f_i \in \mathcal{O}_S^*(U_i)$. Writing $U_i = \Spec(A_i)$ we see there exists a global relative complete intersection $A_i \to B_i = A_i[T]/(T^n - f_i)$ such that $f_i$ maps to an $n$th power in $B_i$. In other words, setting $V_i = \Spec(B_i)$ we obtain a syntomic covering $\mathcal{V} = \{V_i \to S\}_{i \in I}$ and trivializations $\varphi_i : (\mathcal{L}, \alpha)|_{V_i} \to (\mathcal{O}_{V_i}, 1)$. \medskip\noindent We will use this result (the existence of the covering $\mathcal{V}$) to associate to this pair a cohomology class in $H^1_{syntomic}(S, \mu_{n, S})$. We give two (equivalent) constructions. \medskip\noindent First construction: using {\v C}ech cohomology. Over the double overlaps $V_i \times_S V_j$ we have the isomorphism $$ (\mathcal{O}_{V_i \times_S V_j}, 1) \xrightarrow{\text{pr}_0^*\varphi_i^{-1}} (\mathcal{L}|_{V_i \times_S V_j}, \alpha|_{V_i \times_S V_j}) \xrightarrow{\text{pr}_1^*\varphi_j} (\mathcal{O}_{V_i \times_S V_j}, 1) $$ of pairs. By (\ref{equation-isomorphisms-pairs}) this is given by an element $\zeta_{ij} \in \mu_n(V_i \times_S V_j)$. We omit the verification that these $\zeta_{ij}$'s give a $1$-cocycle, i.e., give an element $(\zeta_{i_0i_1}) \in \check C(\mathcal{V}, \mu_n)$ with $d(\zeta_{i_0i_1}) = 0$. Thus its class is an element in $\check H^1(\mathcal{V}, \mu_n)$ and by Theorem \ref{theorem-cech-ss} it maps to a cohomology class in $H^1_{syntomic}(S, \mu_{n, S})$. \medskip\noindent Second construction: Using torsors. Consider the presheaf $$ \mu_n(\mathcal{L}, \alpha) : U \longmapsto \mathit{Isom}_U((\mathcal{O}_U, 1), (\mathcal{L}, \alpha)|_U) $$ on $(\Sch/S)_{syntomic}$. We may view this as a subpresheaf of $\SheafHom_\mathcal{O}(\mathcal{O}, \mathcal{L})$ (internal hom sheaf, see Modules on Sites, Section \ref{sites-modules-section-internal-hom}). Since the conditions defining this subpresheaf are local, we see that it is a sheaf. By (\ref{equation-isomorphisms-pairs}) this sheaf has a free action of the sheaf $\mu_{n, S}$. Hence the only thing we have to check is that it locally has sections. This is true because of the existence of the trivializing cover $\mathcal{V}$. Hence $\mu_n(\mathcal{L}, \alpha)$ is a $\mu_{n, S}$-torsor and by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsors-h1} we obtain a corresponding element of $H^1_{syntomic}(S, \mu_{n, S})$. \medskip\noindent Ok, now we have to still show the following \begin{enumerate} \item The two constructions give the same cohomology class. \item Isomorphic pairs give rise to the same cohomology class. \item The cohomology class of $(\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha')$ is the sum of the cohomology classes of $(\mathcal{L}, \alpha)$ and $(\mathcal{L}', \alpha')$. \item If the cohomology class is trivial, then the pair is trivial. \item Any element of $H^1_{syntomic}(S, \mu_{n, S})$ is the cohomology class of a pair. \end{enumerate} We omit the proof of (1). Part (2) is clear from the second construction, since isomorphic torsors give the same cohomology classes. Part (3) is clear from the first construction, since the resulting {\v C}ech classes add up. Part (4) is clear from the second construction since a torsor is trivial if and only if it has a global section, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-trivial-torsor}. \medskip\noindent Part (5) can be seen as follows (although a direct proof would be preferable). Suppose $\xi \in H^1_{syntomic}(S, \mu_{n, S})$. Then $\xi$ maps to an element $\overline{\xi} \in H^1_{syntomic}(S, \mathbf{G}_{m, S})$ with $n \overline{\xi} = 0$. By Theorem \ref{theorem-picard-group} we see that $\overline{\xi}$ corresponds to an invertible sheaf $\mathcal{L}$ whose $n$th tensor power is isomorphic to $\mathcal{O}_S$. Hence there exists a pair $(\mathcal{L}, \alpha')$ whose cohomology class $\xi'$ has the same image $\overline{\xi'}$ in $H^1_{syntomic}(S, \mathbf{G}_{m, S})$. Thus it suffices to show that $\xi - \xi'$ is the class of a pair. By construction, and the long exact cohomology sequence above, we see that $\xi - \xi' = \partial(f)$ for some $f \in H^0(S, \mathcal{O}_S^*)$. Consider the pair $(\mathcal{O}_S, f)$. We omit the verification that the cohomology class of this pair is $\partial(f)$, which finishes the proof of the first identification (with fppf replaced with syntomic). \medskip\noindent To see the first, note that if $n$ is invertible on $S$, then the covering $\mathcal{V}$ constructed in the first part of the proof is actually an \'etale covering (compare with the proof of Lemma \ref{lemma-kummer-sequence}). The rest of the proof is independent of the topology, apart from the very last argument which uses that the Kummer sequence is exact, i.e., uses Lemma \ref{lemma-kummer-sequence}.