Definition 4

With the hypotheses and notation of Corollary 2 to Proposition 4, a prime ideal $\mathfrak{p}$ of $A$ is said to decompose completely in $K'$ if the number of prime ideals of $A'$ lying above $\mathfrak{p}$ is equal to $[K':K]$.

It amounts to the same to say that, for a prime ideal $\mathfrak{p}'$ of $A'$ lying over $\mathfrak{p}$, the subgroup $\mathscr{G}^{\mathbf{Z}}(\mathfrak{p}')$ is equal to the subgroup $\mathscr{M}$ leaving invariant all the elements of $A'$, or that $A^{\mathbf{Z}}(\mathfrak{p}') = A'$, or that $\mathscr{G}/\mathscr{M}$ operates faithfully on the set of prime ideals of $A'$ lying above $\mathfrak{p}$.