Proposition 4

— Let E be a p-radical extension of finite degree of a field K. Then $[E : K]$ is a power of the characteristic exponent p of K.

Since E is a p-radical extension of finite degree of K, there are elements $a_1, \ldots, a_m$ of E, p-radical over K, such that $E = K(a_1, \ldots, a_m)$. Let i be in the range 1 to m ; since $a_i$ is a fortiori p-radical over $K(a_1, \ldots, a_{i-1})$, the degree

is a power of p (V, p. 24, Prop. 1). We have $[E : K] = n_1 \ldots n_m$ by V, p. 18, Th. 2, whence the result.