Proposition 10

For every Galois extension $L$ of finite degree of $K$, the homomorphism $a_L$ is bijective.

Let $x \in L$ be such that $\mathscr{P}(x) \in K$ and the residue class of $\mathscr{P}(x) \mod \mathscr{P}(K)$ lies in the kernel of $a_L$. For every $\sigma \in \mathrm{Gal}(L/K)$ we have by definition $\sigma(x) = x$; hence $x \in K$ and $\mathscr{P}(x) \in \mathscr{P}(K)$. This proves the injectivity of $a_L$. Now let $f : \mathrm{Gal}(L/K) \to \mathbf{F}_p$ be a homomorphism; for all $\sigma, \tau \in \mathrm{Gal}(L/K)$ we have

By V, p. 65, Cor. 2 there exists $x \in L$ such that $f(u) = \sigma(x) - x$ for all $o \in \mathrm{Gal}(L/K)$. Since $f(o) \in \mathbf{F}_p$ we have $\mathscr{P}(\sigma(x)) = \mathscr{P}(x)$, hence $\sigma(\mathscr{P}(x)) = \mathscr{P}(x)$ for all $a \in \mathrm{Gal}(L/K)$ and $\mathscr{P}(x) \in K$. If $a$ is the residue class of $\mathscr{P}(x) \mod \mathscr{P}(K)$, we have $f(\sigma) = [\sigma, a)$, so $f = a_L(a)$.