Lemma 7

$K(\varphi^{-1}(A))$ is an abelian extension of $K$ of exponent dividing $p$.

Since the polynomials $\varphi - a = X^p - X - a,\ a \in A$ are separable over $K$, the extension $L = K(\varphi^{-1}(A))$ is Galois. Let $a \in \mathrm{Gal}(L/K)$ and $x \in \varphi^{-1}(A)$; we have $\varphi(\sigma(x)) = \varphi(x)$, hence $\sigma(x) - x \in \mathbf{F}_p$, that is, $\sigma(x) = x + i,\ i \in \mathbf{F}_p$. This implies that $\sigma^p(x) = x + pi = x$, hence $u^p = 1$; similarly if $a' \in \mathrm{Gal}(L/K)$ and $\sigma'(x) = x + j$, then we have $\sigma \circ \sigma'(x) = x + i + j = a' \circ \sigma(x)$, so $\sigma \circ \sigma' = \sigma' \circ \sigma$.