Finite purely inseparable rational extensions admit a finite Frobenius envelope

Choose a finite $F$-basis of $L$ and enumerate it as $(u_1,\dots ,u_r)$; by [L2] it is an $F$-spanning set, so every element of $L$ is an $F$-linear combination of the $u_j$, hence lies in the subfield $F(u_1,\dots ,u_r)$ generated by them, while conversely $F(u_1,\dots ,u_r)\subseteq L$; thus $L=F(u_1,\dots ,u_r)$ by [L3].

Evaluation at $x_1,\dots ,x_d$ gives a homomorphism $K[X_1,\dots ,X_d]\to F$ that is injective by [L14], with image the subring generated by $K$ and the $x_i$; since $F$ is the field generated by these elements and contains $K[x_1,\dots ,x_d]$, [L4] gives $F=\mathrm{Frac}(K[x_1,\dots ,x_d])$, the fraction field of the domain $K[x_1,\dots ,x_d]$ of [L5]. In particular a nonzero polynomial in the $x_i$ with coefficients in $K$ is not zero in $F$.

By [L1] applied to each generator $u_j$ of step 1.1 there are $e_j\in \mathbb{N}$ with $u_j^{p^{e_j}}\in F$. If $r\ge 1$ put $e:=\max \{e_1,\dots ,e_r\}$ and otherwise put $e:=0$; set $q:=p^e$ and ${\gamma }_j:=u_j^q\in F$ for $j=1,\dots ,r$.

By step 1.2 each ${\gamma }_j$ is a fraction, so there are $h_j,k_j\in K[x_1,\dots ,x_d]$ with $k_j\mathrm{\ne }0$ and ${\gamma }_j=h_j\mathrm{/}{k}_j$. Let $S\subseteq K$ be the finite set of all coefficients occurring in the polynomials $h_1,k_1,\dots ,h_r,k_r$ and enumerate $S=\{c_1,\dots ,c_s\}$.

Put ${\mathrm{\Omega }}_0:=F$. For $i=1,\dots ,d$ choose, using [L7], a monic irreducible factor $g_i\in {\mathrm{\Omega }}_{i-1}[T]$ of $T^q-x_i$, set ${\mathrm{\Omega }}_i:={\mathrm{\Omega }}_{i-1}[T]\mathrm{/}(g_i)$ and $t_i:=T+(g_i)$; then ${\mathrm{\Omega }}_i$ is a field containing ${\mathrm{\Omega }}_{i-1}$ and $t_i^q=x_i$.

For $i=1,\dots ,s$ choose a monic irreducible factor $g_{d+i}\in {\mathrm{\Omega }}_{d+i-1}[T]$ of $T^q-c_i$, set ${\mathrm{\Omega }}_{d+i}:={\mathrm{\Omega }}_{d+i-1}[T]\mathrm{/}(g_{d+i})$ and ${\alpha }_i:=T+(g_{d+i})$. Then $\mathrm{\Omega }:={\mathrm{\Omega }}_{d+s}$ is a field containing $F$ and ${\alpha }_i^q=c_i$ for every $i$.

The subfield $K^{\mathrm{\prime }}:=K({\alpha }_1,\dots ,{\alpha }_s)$ of $\mathrm{\Omega }$ contains $K$ and is finite over $K$, and every element of $K^{\mathrm{\prime }}$ has its $p^{es}$-th power in $K$: each ${\alpha }_i$ is algebraic over the preceding field with a power basis of length at most $q$ by [L11], so raising an element of $K({\alpha }_1,\dots ,{\alpha }_i)$ to the $p^e$-th power uses [L6], ${\alpha }_i^q=c_i\in K$ and additivity of Frobenius to land in $K({\alpha }_1,\dots ,{\alpha }_{i-1})$, and $s$ such steps land in $K$; the same degree bounds give $[K^{\mathrm{\prime }}:K]\le q^s$ by [L12]. Hence $K^{\mathrm{\prime }}\mathrm{/}K$ is finite purely inseparable by [L1], [L12] and [L13].

Initial embedding: the inclusion ${\phi }_0:F\to \mathrm{\Omega }$, ${\phi }_0(z)=z$, is an injective field homomorphism fixing $F$ pointwise.

Moreover $\mathrm{\Omega }=K^{\mathrm{\prime }}(t_1,\dots ,t_d)$ by [L3], and $\mathrm{\Omega }$ has characteristic $p$ by [L13].

Inductive claim. Let $1\le j\le r$ and suppose that ${\phi }_{j-1}:F_{j-1}\to \mathrm{\Omega }$ is an injective field homomorphism fixing $F$ pointwise, where $F_{j-1}:=F(u_1,\dots ,u_{j-1})$. If $u_j\in F_{j-1}$, then $F_j=F_{j-1}$ and ${\phi }_{j-1}$ itself is the required extension.

In the remaining case $u_j\mathrm{\notin }{F}_{j-1}$ put $D_j:=\{n\in \mathbb{N}:u_j^{p^n}\in F_{j-1}\}$. This set is nonempty because $e_j\in D_j$ by step 2.1, so by [L8] it has a least element $d_j$; here $d_j\ge 1$ because $u_j\mathrm{\notin }{F}_{j-1}$, and $d_j\le e_j\le e$. Put ${\beta }_j:=u_j^{p^{d_j}}\in F_{j-1}$.

In the situation of step 7.1 the element ${\beta }_j$ is not a $p$th power in $F_{j-1}$: if ${\beta }_j=b^p$ with $b\in F_{j-1}$, then $(u_j^{p^{d_j-1}}{)}^p={\beta }_j=b^p$, so injectivity of Frobenius over $F_{j-1}$, available by [L6] and [L13], gives $u_j^{p^{d_j-1}}=b\in F_{j-1}$, contradicting the minimality of $d_j$. Hence $m_j(T):=T^{p^{d_j}}-{\beta }_j$ is monic irreducible over $F_{j-1}$ by [L9], and $m_j(u_j)=0$.

In the situation of step 7.1 define, inside $\mathrm{\Omega }$, the elements $H_j:={\sum }_a{\alpha }_{c_a}{t}^a,K_j:={\sum }_b{\alpha }_{d_b}{t}^b,$ where the sums run over the finitely many exponent vectors $a$ and $b$ occurring in $h_j$ and in $k_j$, with $t^a:=t_1^{a_1}\cdots t_d^{a_d}$ and likewise for $b$; then $H_j,K_j\in \mathrm{\Omega }$.

Frobenius in $\mathrm{\Omega }$, licit by step 6.1 and [L6], gives $H_j^q={\sum }_a{\alpha }_{c_a}^q{t}^{aq}={\sum }_a{c}_a(t^q{)}^a=h_j(x_1,\dots ,x_d)$ and likewise $K_j^q=k_j(x_1,\dots ,x_d)\mathrm{\ne }0$ by step 1.2, so $K_j\mathrm{\ne }0$ and the quotient $W_j:=H_j\mathrm{/}{K}_j\in \mathrm{\Omega }$ is defined and satisfies $W_j^q={\gamma }_j$.

Since ${\beta }_j=u_j^{p^{d_j}}$ and $q=p^e$ with $d_j\le e$ by step 7.1, one has ${\beta }_j^{p^{e-d_j}}=u_j^{p^e}={\gamma }_j$; since ${\phi }_{j-1}$ fixes $F$ pointwise and ${\gamma }_j\in F$ by step 2.1, step 9.1 gives ${\phi }_{j-1}({\beta }_j{)}^{p^{e-d_j}}={\phi }_{j-1}({\beta }_j^{p^{e-d_j}})={\phi }_{j-1}({\gamma }_j)={\gamma }_j=W_j^{p^e}=(W_j^{p^{d_j}}{)}^{p^{e-d_j}},$ and injectivity of the $(e-d_j)$-fold Frobenius power on $\mathrm{\Omega }$ gives ${\phi }_{j-1}({\beta }_j)=W_j^{p^{d_j}}$: that is, $W_j$ is a root in $\mathrm{\Omega }$ of the transported polynomial ${\sigma }_{\ast }(m_j)=T^{p^{d_j}}-{\phi }_{j-1}({\beta }_j)$, where $\sigma :={\phi }_{j-1}$ is regarded as an isomorphism $F_{j-1}\to {\phi }_{j-1}(F_{j-1})$.

Applying [L10] to $\sigma :={\phi }_{j-1}$, to the monic irreducible $m_j\in F_{j-1}[T]$ of step 8.1, to the root $u_j$ of $m_j$ in the extension $F_j\mathrm{/}{F}_{j-1}$, and to the root $W_j$ of ${\sigma }_{\ast }(m_j)$ in the extension $\mathrm{\Omega }\mathrm{/}{\phi }_{j-1}(F_{j-1})$, we obtain a field isomorphism $F_j\to {\phi }_{j-1}(F_{j-1})(W_j)\subseteq \mathrm{\Omega }$ extending ${\phi }_{j-1}$ and sending $u_j\mapsto W_j$; viewed as a map into $\mathrm{\Omega }$ it is an injective field homomorphism ${\phi }_j$ fixing $F$ pointwise.

Steps 5.2, 6.2 and 11.1 give, by induction on $j=0,1,\dots ,r$, injective field homomorphisms ${\phi }_j:F(u_1,\dots ,u_j)\to \mathrm{\Omega }$ fixing $F$ pointwise; in particular ${\phi }_r:L\to \mathrm{\Omega }$ is an embedding of $L$ over $F$.

By step 6.1 the field $\mathrm{\Omega }$ equals $K^{\mathrm{\prime }}(t_1,\dots ,t_d)$ with $t_i^q=x_i$; writing $x_i^{1\mathrm{/}q}:=t_i$, this subfield is $K^{\mathrm{\prime }}(x_1^{1\mathrm{/}q},\dots ,x_d^{1\mathrm{/}q})$, and $K^{\mathrm{\prime }}\mathrm{/}K$ is finite purely inseparable by step 5.1. Together with step 12.1 this exhibits the required embedding of $L$ over $F$ into $K^{\mathrm{\prime }}(x_1^{1\mathrm{/}q},\dots ,x_d^{1\mathrm{/}q})$, so the lemma is proved.