Proposition 4

— Let $x = (x_i)_{i \in I}$ be a p-basis of K. For every integer $n \geq 1$, the family $(x^u)_{u \in \Lambda(n)}$ is a basis of K over $K^{p^n}$.

For $n = 1$ the assertion reduces to Prop. 1 (V, p. 98). The set $\Lambda(n)$ consists of elements of $N^{(I)}$ of the form $a = \beta + p^{n-1}\gamma$ with $\beta \in \Lambda(n-1)$ and $\gamma \in \Lambda(1)$. Such a decomposition is unique and we have $x^\alpha = x^\beta (x^\gamma)^{p^{n-1}}$. Moreover, the family $(x_i^{p^{n-1}})_{i \in I}$ is clearly a p-basis of $K^{p^{n-1}}$ over $K^{p^n}$, hence the family $(x^\gamma)^{p^{n-1}}$ is a basis of $K^{p^{n-1}}$ over $K^{p^n}$. We thus obtain the conclusion by induction, using II, p. 222, Prop. 25.