Lemma 8

Let L be a Galois extension of K. There exists a unique mapping $(\sigma, a) \mapsto [a, a')$ of $\mathrm{Gal}(L/K) \times ((\varphi(L) \cap K)/\varphi(K))$ into $\mathbf{F}_p$ such that for all $\sigma \in \mathrm{Gal}(L/K)$ and every element $x \in L$ such that $\varphi(x) \in K$ we have, on denoting by $\overline{\varphi(x)}$ the residue class of $\varphi(x)$ mod $\varphi(K)$,

This mapping is Z-bilinear (for $\sigma, \tau \in \mathrm{Gal}(L/K)$, $a, b \in (\mathscr{P}(L) \cap K)/\mathscr{P}(K)$, we have $[\sigma \tau, a) = [\sigma, a) + [\tau, a)$, $[\sigma, a + b) = [\sigma, a) + [a, b)$).

For the right-hand side of (15) is an element of $\mathbf{F}_p$ which depends only on the residue class of $\mathscr{P}(x) \mod \mathscr{P}(K)$; this proves the first assertion; the second may be verified without difficulty.

For every Galois extension $L$ of $K$ let us write

for the homomorphisms obtained from the above Z-bilinear mapping (V, p. 87).