Corollary

— Let L' be an extension of a field K, generated by two linearly disjoint subextensions K' and L. Suppose that L is p-radical of height $\leq 1$ over K; then L' is p-radical of height $\leq 1$ over K'. Moreover, for a family of elements of L to be p-free over K (resp. a p-basis of $L$ over K) it is necessary and sufficient that it should be p-free over K' (resp. a p-basis of L' over K').

We have ${L'}^p \subset K \subset K'$ and $L' = K'(L)$, whence ${L'}^p = {K'}^p({L'}^p) \subset K'$. In other words, L' is a p-radical extension of height $\leq 1$ of K'. The other assertions of the corollary follow at once from Prop. 1 and from V, p. 14, Prop. 5.