Exercise 1, § 13

Let $K$ be a field of characteristic $p > 0$ and $E$ an extension of $K$; $E$ is a $p$-radical extension of height $\leq 1$ of $K(E^p)$. The cardinal of a $p$-basis of $E$ over $K(E^p)$ (cf. V, p. 103, Th. 2) is called the degree of imperfection of $E$ over $K$. If $K(E^p) = E$, $E$ is said to be relatively perfect over $K$; if $E$ is perfect, it is relatively perfect over each of its subfields. The (absolute) degree of imperfection of $K$ is the degree of imperfection of $K$ over its prime subfield $F_p$, or also the cardinal of a (absolute) $p$-basis of $K$. a) If $B$ is a $p$-basis of $E$ over $K(E^p)$, show that for every integer $k > 0$ we have $E = K(E^{p^k})(B)$. b) Suppose that $E \subset K^{p^{-n}}$ for an integer $n > 0$. Show that for the degree $[E : K]$ to be finite it is necessary and sufficient that the degree of imperfection $m_0$ of $E$ over $K$ should be finite; then $m_0$ is the least cardinal of the sets $S$ such that $E = K(S)$. Show that if $m_k$ is the degree of imperfection of $K(E^{p^k})$ over $K$, then $m_{k+1} \leq m_k$ for all $k$, and if $f = \sum_k m_k$, then $[E : K] = p^f$.