Separable generation after finite purely inseparable extensions

Reduction and set-up. If the characteristic is $0$, then $k$ is perfect by [F19], so [F20] makes $K\mathrm{/}k$ separably generated and $k^{\mathrm{\prime }}=k$, $K^{\mathrm{\prime }}=K$ are finite purely inseparable over their bases by [F8] (in characteristic $0$ the only purely inseparable extension is the trivial one). Assume from now on that the characteristic is $p>0$. Fix a finite generating set ${\alpha }_1,\dots ,{\alpha }_n$ of $K$ over $k$ by [F21] and choose a maximal algebraically independent subset $\{x_1,\dots ,x_r\}$ of it, which is a transcendence basis of $K\mathrm{/}k$ by [F3, F2]. Put $F_0:=k(x_1,\dots ,x_r)$; every ${\alpha }_i$ is algebraic over $F_0$ by maximality, so $K\mathrm{/}{F}_0$ is finitely generated algebraic, hence finite by [F4]. Let $K_s$ be the separable closure of $F_0$ in $K$; then $K_s\mathrm{/}{F}_0$ is finite separable by [F6, F4] and $K\mathrm{/}{K}_s$ is purely inseparable by [F7], finite by [F18], so that $d:={\log }_p[K:K_s]$ is a nonnegative integer by [F9, F8].

An element of $K$ with a $p$-th root of the right shape. Assume $d\ge 1$, so $K\mathrm{\ne }{K}_s$. By [F8] and [F7] applied to an element of $K\setminus K_s$ there is ${\beta }_0\in K$ and a minimal $e\ge 1$ with ${\beta }_0^{p^e}\in K_s$; then $\beta :={\beta }_0^{p^{e-1}}$ satisfies $\beta \mathrm{\notin }{K}_s$ by minimality of $e$ and $\alpha :={\beta }^p\in K_s$. Since $\alpha \in K_s$ is separable over $F_0$ by [F5, F6], its minimal polynomial $P\in F_0[T]$ over $F_0$ is separable by [F12, F5], hence $\gcd (P,P^{\mathrm{\prime }})=1$ by [F13] and $P$ has pairwise distinct roots; write $P={\sum }_i{a}_i{T}^i$ with $a_i\in F_0=k(x_1,\dots ,x_r)$.

A finite purely inseparable base change making the coefficients $p$-th powers. Write each $a_i=f_i\mathrm{/}{g}_i$ with $f_i,g_i\in k[x_1,\dots ,x_r]$ and $g_i\mathrm{\ne }0$, and let $C\subseteq k$ be the finite set of all coefficients occurring in the finitely many polynomials $f_i,g_i$. Choose an algebraic closure $\mathrm{\Omega }$ of $K$ and inside it put $k^{\mathrm{\prime }}:=k(c^{1\mathrm{/}p}:c\in C)$, so that $k^{\mathrm{\prime }}\mathrm{/}k$ is finite purely inseparable by [F17, F8]; put $F:=k^{\mathrm{\prime }}(x_1^{1\mathrm{/}p},\dots ,x_r^{1\mathrm{/}p})\subseteq \mathrm{\Omega }$, the compositum of $k^{\mathrm{\prime }}$ and $F_0$. Each monomial $x^m$ with $m\in {\mathbb{N}}^r$ is a $p$-th power in $F$: writing $m=p{m}^{\mathrm{\prime }}+m^{\mathrm{\prime }\mathrm{\prime }}$ with $m^{\mathrm{\prime }\mathrm{\prime }}\in \{0,\dots ,p-1{\}}^r$ one has $x^m=(x^{m^{\mathrm{\prime }}}(x^{1\mathrm{/}p}{)}^{m^{\mathrm{\prime }\mathrm{\prime }}}{)}^p$. Each $c\in C$ equals $(c^{1\mathrm{/}p}{)}^p$, so every $f_i$ and $g_i$ is a finite sum of $p$-th powers, hence a $p$-th power by [F16], and therefore each $a_i=f_i\mathrm{/}{g}_i$ is a $p$-th power in $F$. Write $a_i=c_i^p$ with $c_i\in F$ and put $R:={\sum }_i{c}_i{T}^i\in F[T]$.

The $p$-th root of $\alpha$ is separable. By [F15] and [F16], $P(T^p)={\sum }_i{a}_i{T}^{pi}={\sum }_i{c}_i^p{T}^{pi}=({\sum }_i{c}_i{T}^i{)}^p=R(T{)}^p$ in $F[T]$. Let ${\alpha }_1,\dots ,{\alpha }_n\in \mathrm{\Omega }$ be the distinct roots of $P$ (so $n=\mathrm{deg}P=\mathrm{deg}R$, and $n\ge 1$), and for each $j$ choose ${\gamma }_j\in \mathrm{\Omega }$ with ${\gamma }_j^p={\alpha }_j$, which exists because $\mathrm{\Omega }$ is algebraically closed. Then $R({\gamma }_j{)}^p=P({\gamma }_j^p)=P({\alpha }_j)=0$, so $R({\gamma }_j)=0$, and the ${\gamma }_j$ are distinct because ${\gamma }_j^p={\alpha }_j$ are. Hence $R$ has $\mathrm{deg}R$ distinct roots in $\mathrm{\Omega }$, so $R$ is separable over $F$ by [F12] and [F13] applied to its distinct-root factorisation. Since $\alpha ={\beta }^p$ is a root of $P$, the element $\beta$ satisfies $R(\beta {)}^p=P({\beta }^p)=P(\alpha )=0$, hence $R(\beta )=0$; and $\beta \in L:=K\cdot F\subseteq \mathrm{\Omega }$, a compositum which is finitely generated over $k^{\mathrm{\prime }}$. As $R$ is separable with the root $\beta$, the minimal polynomial of $\beta$ over $F$ divides $R$ by [F14] and is separable, so $\beta$ is separable over $F$ by [F5] and lies in the separable closure $L_s$ of $F$ in $L$ by [F6].

The case $d=0$. If $d=0$ then $[K:K_s]=1$, so $K=K_s$ and $K\mathrm{/}{F}_0$ is finite separable by [F6]; then $x_1,\dots ,x_r$ is a separating transcendence basis of $K\mathrm{/}k$ by [F1], and with $k^{\mathrm{\prime }}=k$, $K^{\mathrm{\prime }}=K$ the conclusion holds trivially.

The separable closure of $F$ in $L$ and the degree drop. $L\mathrm{/}K$ is finite purely inseparable and $F\mathrm{/}{F}_0$ is purely inseparable. First, $K_{s}F\mathrm{/}F$ is separable: $K_s\mathrm{/}{F}_0$ is finite separable by step 1.1, and a compositum of a separable algebraic extension with a further extension is separable because the minimal polynomial over the larger field divides the separable minimal polynomial over the smaller one by [F14]. Second, $L\mathrm{/}{K}_{s}F$ is purely inseparable by [F17]. Hence $L_s=K_{s}F$: one inclusion holds because $K_{s}F\mathrm{/}F$ is separable and $L_s$ is the largest separable intermediate field by [F6], and for the other, an element $\gamma \in L_s$ is separable over $F$ and has ${\gamma }^{p^e}\in K_{s}F$ for some $e\ge 0$ by [F8], so it is simultaneously separable and purely inseparable over $K_{s}F$ and therefore already lies in $K_{s}F$. Since $\beta \in L_s$ and $\beta \mathrm{\notin }{K}_s$, the field $K_s(\beta )$ satisfies $[K_s(\beta ):K_s]=p$: the polynomial $T^p-\alpha \in K_s[T]$ vanishes at $\beta$ while $\alpha$ is not a $p$-th power in $K_s$ (a relation $\alpha ={\gamma }^p$ would give $(\beta \mathrm{/}\gamma {)}^p=1$, hence $\beta =\gamma$ by [F16]), so $T^p-\alpha$ is irreducible by [F11] and is the minimal polynomial of $\beta$ over $K_s$ by [F14]. Put $E:=K\cap L_s$, an intermediate field of $K\mathrm{/}{K}_s$ containing $K_s(\beta )$, and choose ${\theta }_1,\dots ,{\theta }_m\in K$ with $K=E({\theta }_1,\dots ,{\theta }_m)$, possible since $K\mathrm{/}E$ is finite by [F18]. Then $L=L_s\cdot K$ equals $L_s({\theta }_1,\dots ,{\theta }_m)$, and for each $i$ the minimal polynomial of ${\theta }_i$ over $L_s({\theta }_1,\dots ,{\theta }_{i-1})$ divides its minimal polynomial over $E({\theta }_1,\dots ,{\theta }_{i-1})$ by [F14], so those two extensions satisfy the degree inequality $[L_s({\theta }_1,\dots ,{\theta }_i):L_s({\theta }_1,\dots ,{\theta }_{i-1})]\le [E({\theta }_1,\dots ,{\theta }_i):E({\theta }_1,\dots ,{\theta }_{i-1})]$; multiplying over $i$ and applying [F18] twice yields $[L:L_s]\le [K:E]\le [K:K_s(\beta )]=[K:K_s]\mathrm{/}p$, the last equality being [F18] for the tower $K_s\subseteq K_s(\beta )\subseteq K$.

Induction. The extension $L\mathrm{/}{k}^{\mathrm{\prime }}$ is finitely generated by [F21] and has characteristic $p$, and by step 2.2 its invariant ${\log }_p[L:L_s]$ is strictly smaller than $d={\log }_p[K:K_s]$. Applying the induction hypothesis (on the nonnegative integer $d$, with the same statement for the pair $L\mathrm{/}{k}^{\mathrm{\prime }}$) produces finite purely inseparable extensions $k^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{k}^{\mathrm{\prime }}$ and $L^{\mathrm{\prime }}\mathrm{/}L$ with $L^{\mathrm{\prime }}\mathrm{/}{k}^{\mathrm{\prime }\mathrm{\prime }}$ separably generated. Then $k^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}k$ is finite purely inseparable by [F17, F9] and $L^{\mathrm{\prime }}\mathrm{/}K$ is finite purely inseparable by [F17] since $L\mathrm{/}K$ is, so $k^{\mathrm{\prime }\mathrm{\prime }}$ and $L^{\mathrm{\prime }}$ satisfy the conclusion for $K\mathrm{/}k$. The base case $d=0$ of the induction is step 2.1, so the assertion holds for every $d\ge 0$.

Conclusion. In characteristic $p>0$ steps 1.2–1.4, 2.1, 2.2 and 3.1 produce the required finite purely inseparable $k^{\mathrm{\prime }}\mathrm{/}k$ and $K^{\mathrm{\prime }}\mathrm{/}K$ with $K^{\mathrm{\prime }}\mathrm{/}{k}^{\mathrm{\prime }}$ separably generated, the induction being on the integer $d={\log }_p[K:K_s]$ of step 1.1; the characteristic $0$ case is step 1.1. ∎