Exercise 7, § 16

Assume that the hypotheses of Ex. 6 hold, and further that the degree of imperfection of $E$ over $K$ (V, p. 170, Ex. 1) is finite (which is always the case if $E$ is a finitely generated extension of $K$).

a) Let $B$ be a p-basis of $F$ over $K(F^p)$ which is also a separating transcendence basis of $F$ over $K$ (and a transcendence basis of $E$ over $K$). Show that for every integer $k > 0$, $B^{p^k}$ is a p-independent subset of $K(E^{p^k})$ over $K(E^{p^{k+1}})$. Let $q = \operatorname{tr.deg}_K E$; if $m_k + q$ is the degree of imperfection of $K(E^{p^k})$ over $K(E^{p^{k-1}})$, show that $p^f$, where $f = \sum_{k=0}^{h-1} m_k$, is equal to the degree $[E : F]$ and is therefore independent of the choice of the distinguished subextension $F$ of $E$. The integer $p^f$ is called the order of inseparability of $E$ over $K$ and is written $[E : K]_i$; this number coincides with the inseparable degree denoted by $[E : K]$, in V, p. 46 when $E$ is of finite degree.

b) For $0 \leq k \leq h - 1$, $[K^{p^{-k}}(E) : K^{p^{-k}}]$ is equal to $p^{f_k}$, where $f_k = \sum_{j=k}^{h-1} m_j$.

c) Let $L$ be a subextension of $E$, separable over $K$ and such that $E$ is p-radical and of finite degree over $L$; show that $[E : L] \geq [E : K]$, (if $[L(E^{p^k}) : K(E^{p^k})] = p^{r_k}$, show that $r_{k+1} \leq r_k + q$, by considering a basis of $L(E^{p^k})$ over $K(E^{p^k})$ and a basis of $L$ over $K(L^p)$).

d) Let $K'$ be an extension of $K$ such that $K'$ and $E$ are contained in the same algebraically closed extension $\Omega$ of $K$. Show that if $K'(F)$ is separable over $K'$, it is a distinguished subextension of the extension $K'(E)$ of $K'$. This is so when $K'$ and $E$ are algebraically disjoint over $K$ and then we have $[K'(E) : K'] \leq [E : K]$,; equality holds when $K'$ and $E$ are linearly disjoint over $K$ or when $K'$ is a separable algebraic extension of $K$.

e) Show that $[E : K]_i$ is the least value of $[E : K(B)]_i$ for all transcendence bases $B$ of $E$ over $K$; if $K$ is the relative algebraic closure of $K$ in $\Omega$ (which is an algebraic closure of $K$), then $[E : K(B)]_i = [E : K]_i [\bar{K}(E) : \bar{K}(B)]_i$. If $L$ is a subextension of $E$ such that $E$ is an algebraic extension of $L$ and $\bar{K}(E)$ a separable (algebraic) extension of $\bar{K}(L)$, then the largest separable extension $F$ of $L$ contained in $E$ is distinguished.

f) Suppose that $E$ is a finitely generated extension of $K$; show that for every subextension $L$ of $E$ we have $[L : K]_i \leq [E : K]_i \leq [L : K]_i [E : L]_i$. Give an example where