Proposition 4

Let p be the characteristic exponent of K and E an extension of finite degree of K. Let $\sigma_1, \ldots, \sigma_n$ be the distinct K-homomorphisms of E into an algebraic closure $\Omega$ of K, and let $p^e = [E:K]_i$. For each $x \in E$ we have in $\Omega$

Let E, be the relative separable closure of K in E; then $E_s$ is a separable extension of degree $n$ of K and $\sigma_1, \ldots, \sigma_n$ induce distinct K-homomorphisms of E, into R; further, E is ap-radical extension of E, of degreep$^e$ ($V$, p. 44, Prop. 13 and p. 46). So Prop. 4 follows from the Formulae (6), (7), (13), (14) and (12).